The Aruch Hashulchan 234:1 says he should skip ashrei. He doesn't make a distinction between coming right when they start or in the middle. Obviously, you cannot be mafsik during kaddish to say ashrei. (I would have thought to say a few pesukim kedei laamod bitfilla mitoch divrei Torah, but I guess not.)
There is no takanah of saying ashrei after Shmona Esrei (although it is praiseworthy to say ashrei 3x daily).
If you come in the middle of the tzibbur davening, you should start with them only if you will complete the last bracha and step back in time to answer kedusha, ha-el hakadosh, shomea tefilla and kadish shalem. An alternative is to daven word for word saying these divrei kedusha with the shliach tzibbur. These halachos are in Shulchan Aruch 109.
The Aruch Hashulchan allows you to daven a long tefilla even if you miss the above if you started with the tzibbur. This seems to be the minhag.