Can I prevent my tub faucet from leaking by diverting the water to the shower, and shutting off the shower head?
Diverting the leak up to the shower head using the diverter valve should not be a problem, but your assumption that fixing it will be "pretty easy" may not prove to be true. You are correct in that hot and cold supply to the tub/shower valve needs to be shut off. Sometimes apartment shower valves have built in shut-off valves called integral stops.



Sometimes apartments are equipped with individual remote shut-offs. Sometimes the whole building will need to be shut down.
Removing the valves may require special tools like deep valve sockets.



The repair may be simple washer replacement or may require more complicated cartridge replacement. 

I do recommend that you try it but make sure your landlord is cool with it in case something breaks, and have back up shower facilities available in case it ends up being out of service for a while.