This is a fairly large project. Bathtubs, if older, are usually made
from ultra heavy cast iron which is a pretty big chore to remove.
Putting in a shower stall is pretty straight forward if you buy a
pre-assembled white fiberglass unit. If a 36" or 32" square doesn't
fit though, you're gonna have to build a custom shower stall with tile,
plastic, mortar and a LOT of time.
These things are all something a very determined do-it-yourselfer could
accomplish, but you are renting after all, and such things as bathroom
remodels are usually in the interest of the landlord as they increase
the amount they can rent the place for in the future.
I really don't understand why you want to take out the tub and put in a
shower stall though. You mentioned arthritis, but I thought submersing
oneself in warm water was soothing for such a condition.
Correct me where I went wrong.