What kind of insulation is best for apartments?

foam would be the easiest. you would only have to cut a small hole at the top and a small hole in the bottom of the wall between each stud bay, and spray in the foam. then just patch and paintt the holes. fibergl batts you would have to remove most of the drywall and replace. the spray foam is more expensive for the insulation itself but in your case if you add in the cost new drywall then the foam will actually be cheaper.