Repairing Concrete Steps
Thta is the responsibility of the landlord or owner. Tenants can't make repairs or improvements without approval.
That said, paint only covers up the problems, it does not fix them. I am not sure how bad that bottom slab is, but surface fixes rarely last long. It probably needs to dug up and replaced. There is a product called Top'N'Bond that is one of the better cement patching products. But no guarantees it will last, especially if you are in a cold climate with a lot of freeze/thaw cycles.