Solid door on top of hollow apartment door to reduce outside noise?
I doubt if adding a smaller door would make any difference. 

In “sound control”, you want to attack the weak link. That’s why you’re suggesting fixing the door problem rather than improving the sound control in the wall. 

Too bad you can’t temporarily replace the hollow core door with a solid core door, but perhaps 1) adding a screen door, 2) adding weatherstripping, 3) adding insulated panels to the inside to cover the entire doorway, etc. will help.