How do I patch up this cracked corner (plaster walls)?
Caulk.

But before you do that, ask yourself what the reason for the crack may be, and if it's a symptom of something more sinister.   Dry rot?  Termites?  Foundation shifting?  Someone with anger management problems slamming the closet doors?

Of course it may be nothing, in which case you can either patch the entire wall from trim to trim, or just dig out the crack, remove as much dust as possible, paint with white glue to seal the surface and put in several layers of common painter's caulk over several days.  I don't recommend plaster patch in this case, because this seam is likely to move again.  You want something with some flex.

If the plaster is starting to separate from the walls, then Bill Wally has your answer over at : https://www.plastermagic.com/FAQs.html

If the crack opens up again, use a more flexible caulk (not just painter's caulk).