I have a cylinder fuse box in my apartment. Can I run a 220 to hook up a dryer off it?

It has to be a double fuse type for 220 volt (2 hots) assuming your dryer is a 3 wire type. A 4 wire needs a neutral and a ground. Some people just have the ground double as a neutral if it is bonded in the main breaker box but that is not safe though. The dryer needs to be on a dedicated circuit for safety. Not a good idea to 'borrow' power from a fuse box unless there is no other wiring using it.

You would have to use the cartridge fuses if they were available. You cannot use the glass fuses.

If this is an apartment you are renting from someone you have NO choice but to have the landlord do this for you.
Do not mess with the electric in a place you are renting.