Can a FWD gearbox work in a Mid-Engine car?

I have a 1995 Honda Beat PP1 and I want to swap in a 5-speed from a Civic or Integra and I was wondering if a gearbox made for a FWD car would hook up correctly with a Mid-Engine car like mine.

Not really. A FWD car has its engine right over the wheels so that the alignement between the engine and gearbox with are placed horizontally within the engine bay is quite easy to do. In a mid engine car the engine and gearbox are in front of the rear wheels meaning you will need a longitudinally mounted engine and gearbox. The best you could do is a rear engine like a porsche 911 which is just like a FWD but in reverse. Keep in mind a FWD car driving in reverse has the same characteristics as a rear engined, RWD car but you only get the 1 "forward" gear, the steering is at the wrong end and your aerodynamics are all messed up.

in some special cars like the Porsche Cayman, Lamborghini Miura and others the mid engine is actually mounted in a transverse manner which might make the alignment easier but still not just a "quick" bolt on job. Your Beat PP1 has a transversely mounted engine

I've made a picture gallery showing some engine mounting options and why your idea won't really work.

It will if you move the front wheels back about 10 feet so they match up with the transmissions output shafts