I just moved into a new apartment not realizing the back steps were in such disrepair. My knowledge of home repair is limited and I'm on a budget. I'd really appreciate some advice on repairing these steps and then painting them.
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.
you don't have improvement nor repair rights if a tenant except by express agreement & permission,,, that said, you could try polymer-modified repair mtls which're NOT avail at apron stores but would be vastly more permanent than top 'n' bond.
no paint adheres permanently to conc altho there're coatings that we use & have been in place for over 20yrs w/o any degradation.