Not a good idea to fool around with the plumbing in an apartment you do not own. What if you screw it up?
As far as the answer to your question, it is possible you have to shut down the whole building to service the shower. But if it was well designed, there may be screwdriver stops behind the chrome trim flange or flanges. But they may be iffy given old age, and messing with them can get you into a "no-return" situation. It is not your fault the shower is dripping, and your landlord should be happy to fix it for you, especially if he is paying for the wasted water and energy to heat it.