Our sun will someday (in billions of years) go nova, expanding in size and changing color. Anyone nearby is sure to have a bad day. Should this event occur earlier than predicted, however, would Superman be able to intervene? The very condition his powers rely on would also be one of the qualities that would change.
I do not ask what specific actions he might take if his powers were to remain, but rather (assuming any of them would be relevant) would he have them to use at all?
tl;dr: The "event" you're talking about (which is not called a nova) lasts about 2 billion years. During much of that, Superman will still have his yellow-star-given powers. Unfortunately, the only thing he can really do at that point is fly off to find a new solar system before his immortality runs out.
The process of a yellow dwarf star growing and "cooling" into a red giant (which is not a nova, as many others have pointed out) is a slow process. When the hydrogen in its core runs out, the Sun will not immediately get big and red; in fact, it will get hotter , though not significantly hot enough to change its emission spectrum to blue.
The added heat will cause the hydrogen in the shell to fuse, heating it up further and causing it to expand. Eventually the helium -> carbon fusion will start up in the core, at which point the expansion along the red giant branch stops. The star stops being a red giant and starts a rapid descent into white-dwarfiness.
For our sun, the expectation is that it will spend over 2 billion years growing from a yellow dwarf to a red giant, during which time it will not expand very much at all. Over the last 200 million years or so, there will finally be a rapid expansion up to the maximum size of a solar-mass red giant. The change in color is due entirely to the expansion - the total heat of the star is spread out over a larger surface area, which shifts the emission spectrum away from the blue and into the orange-red. But at the beginning of that process, the sun will continue to emit light across the entire spectrum, including yellow.
If we assume that Superman's power comes from the absorption of radiation from a yellow dwarf star (the only kind of "yellow sun" that we know of), then he will continue to have those powers for some time after the hydrogen runs out. We've also seen that Superman doesn't always need direct yellow sunlight to maintain his powers; later comics have depicted the process more like a solar battery. As long as he has "regular" refreshes of yellow light, his powers stick around through, for example, the night time, long-term indoor confinement, trips through space, etc.
If he waits too long and the star begins to ascend along the red giant branch, his powers will diminish until he's back to "normal". As the yellow part of the emission spectrum fades out, presumably Superman will gain less and less benefit from it. What exactly this means has never been explored that I know of; it might make him proportionally weaker, it might take longer for him to "recharge", it might mean he needs to remain outdoors in direct sunlight for longer periods, etc.
Of course, by that time, most predictions say the Earth will have long since been swallowed up by the sun, which will inflate until it reaches into the orbit of Mars. At the least, it will have been rendered uninhabitable, so most likely Superman will have already found a new home.
I'm not sure what he can really do to "prevent" our Sun from going Red-Hulking out. Unlike novae or Ia supernovae, which depend on degenerate stars accumulating more mass until they explode, the red giant phase of the Sun is going to happen as long as it retains the same mass it has now. The only way to stop it, as @Chris Behren's mentions, is to make sure it never runs out of hydrogen.
Unfortunately, if you try to prevent the inevitable by shoving more hydrogen into the core, you increase the mass, which has two effects. First, it shortens the lifetime of the star dramatically (a star 1/10 the mass of the sun lives about 1000 times longer), meaning you have to keep adding more mass more quickly to keep it stable. More importantly, you will eventually cross the threshold from a G-class yellowish-white star into a B-class blue star, thus defeating the purpose entirely :)
It should be point out that our Sun is not yellow; it's white. It appears yellow in our sky because all the blue light has been scattered by our atmosphere. If Superman needs an actualy yellow dwarf star to obtain his powers, then a star even 40% larger would be no good, as we'd pass into the blueish-white A-type star. If he only needs some yellow-wavelength emissions, he would be safe a bit longer, until the B-class and O-class stars (most of which are giant or supergiants and would thus destroy the Earth anyway.)
In real life, our sun isn't of the right solar mass to go nova, just to get that out of the way. According to our best understanding, it will expand to become a red giant, expel it's lighter elements in a non-explosive fashion, and then cool over million of years to become a black dwarf. But this is fiction, and we can still address the question of a nova.
You've got two questions rolled up here - how do you delay a nova, and can Superman do it? I think the first one is the more difficult question. A nova occurs when a star has "burned" up its nuclear fuel, become a white dwarf, and then hydrogen accretes on the surface. I would generally say that by that point, it's too late...you're going to get a nova. So the real question is, how does one delay a star becoming a white dwarf? By restoring a "healthy" balance of hydrogen fuel in the star.
There are two ways to do this: introduce a truly mind-boggling amount of hydrogen mass into the star, or fission helium back into hydrogen by introducing a mind-boggling large amount of energy. I would tend towards the second, as I think that changing the mass of a star is probably a dicey proposition, particularly with stellar convection currents...you might actually accelerate the process inadvertently.
I would say that both of these approaches are beyond Supes' physical abilities. If he wanted to do this, it would be a matter of motivating an entire extremely advanced civilization to undertake a huge engineering project. His role would be much more of a matter of diplomacy and representation of Earth than physical strength.
Lastly, though, if there were a physical aspect that Supes could bring to bear, I would say it would be flying into the heart of the sun and introducing said fission directly with his heat vision. I think that's pretty much God-like in aspect.
No, Superman from most of the DC Universe continuities readers are familiar with would have no chance to effectively stop a star the size of our sun from going nova. Despite his amazing powers, it is not a matter of power, it is a matter of scale that would defeat Superman in such a titanic struggle to control one of the most powerful forces in nature, the death of a star.
Superman derives his powers in an as yet undisclosed manner from the G-type star our planet currently resides around. Since the mechanic for how his power functions has never been clearly defined let’s set this up as a purely physical challenge.
The sun is about to go nova, increasing its energy output exponentially and will engulf the Earth in a matter of months. Can Superman’s powers give him any means of stopping it?
To hazard a guess, I would say no. Yes, Superman’s powers are prodigious allowing him the capabilities to do many things only capable with an advanced technology. But even his most amazing abilities would mean nothing in comparison to the object from which he derives his powers. Our sun is a massive energy source, even a mere solar flare, a star’s least expression of its incredible output possess the power of a 160 billion megatons of TNT in one moment; sufficient enough to give even a Superman pause.
A solar flare is a sudden brightening observed over the Sun's surface or the solar limb, which is interpreted as a large energy release of up to 6 × 10^25 joules of energy (about a sixth of the total energy output of the Sun each second or 160,000,000,000 megatons of TNT equivalent, over 25,000 times more energy than released from the impact of Comet Shoemaker–Levy 9 with Jupiter). Wikipedia - Solar Flare
Superman’s abilities are a function of the energy of the sun being catalyzed though his Kryptonian physiology. So unless his physiology allows him to create an energy output greater than his input (which theoretically he already does) this is not a problem he can tackle. It is a matter of scale.
Assuming he could reach the speed of light, circling the Earth one time at light speed takes him 0.133 seconds. Circling the sun at light speed would take him 14.5 seconds. At our scale he would be extraordinary, at the scale of our sun, he would be just too slow, too small, take too long to do anything. It’s not a slight against him or his powers, it’s just showing size does matter when you leave the planetary scale.
His powers while they are capable of doing things on a planetary scale which would be considered mindboggling, at a solar scale, he is too small to do anything effective enough to stop a star from going nova; simply because the size of the sun, the area it covers, the energy output would simply be far greater than even his organic form, adapted for the processing and conversion of solar energy into super-energetic feats could handle.
Think of Superman in this instance acting more like a fuse than a battery. He would absorb all the energy he could store and even if he could release that energy, just as fast as he could absorb it, he would eventually begin to feel stresses which should eventually overwhelm both his invulnerability and his ability to process, or manipulate energy and begin to tear him apart.
This would be the modern Superman as we know him. We have seen potential futures where Superman Prime becomes so invulnerable he eventually elects to live in the sun itself using his powers in a feedback loop to prevent him from being destroyed by the sun. But Superman Prime is also thousands of years older and evidently an order of magnitude more powerful than any iteration of Superman ever known.
The alternative timeline Superman Prime; a being so powerful he was capable of living inside the sun and beaming his extra converted solar power to his super-powered descendants.
IMO: The comic book version of Superman would be able to fly back in time until he had enough time to work out a way to save humankind, but he probably wouldn't be able to stop the sun from running its course.
