Despite the new 3d visuals, C3 is still a 2D engine. As far as the engine is concerned, everything is actually still on the 'floor' despite how it may look visually.
The built in bullet behavior has no z axis component. You'll need to manually adjust the bullet object's z elevation every tick based on the vertical angle of the camera when it was created. You'll also need to modify the base speed as well, as a bullet shot straight up or down actually isn't "moving" at all.
Anyways, you're going to have trouble trying to make an fps like that since there are no collisions in 3d space anyway. You might be able to fake it by comparing the zposition and zhight of objects while overlapping, but that's going to result in other problems. I imagine you might as well spend your effort learning how to use an actual 3d engine instead, or read up on how verticality in Doom 2 worked, or just go with a completely "flat" levels like Wolfenstein.