It looks like the drain piece is an ABS piece that is glued directly to the drain pipe. Unfortunately, if you don't have access under the tub, it'll probably be difficult to replace. You'll probably need to call a plumber and coordinate with your downstairs neighbors to get that fixed.
In the mean time, yeah throw some caulk or something on the crack to keep water from seeping down. I don't recommend JB Weld, as it will probably be more difficult to remove when you want to fix it properly.