The exterior walls (not the party walls shared with adjoining units)
are usually cold in winter and very warm in summer (upstate NY).
There is insulation, but the unit only has 2x4 walls. I was
wondering... is there a relatively painless way to increase the wall
thickness? Is it worth it? What are the issues with doing this?

I remodelled a house once by firring the 2x4 walls out by nailing a
2x2 on top of each stud. Worked great, and yes it was worth it. It
yielded a stud cavity of 5" instead of 5 1/2, so I did lose a little R
value there, but the 2x2 is a painless way to add some thickness. If
I was doing it again, however, I'd add a layer of rigid insulation
instead of firring the studs. You will have to extend your window
jambs, move electrical boxes, potentially move radiators (if any),
etc.