How to bulk up in body building but not sacrifice speed in fighting?
By training for athletic performance and not bodybuilding.
you'd be able to put on some muscle, but every pound of muscle would have to be very functional and perform well otherwise it's just a weight dragging you down.
id suggest lifting heavy (1-5 reps for 3-5 sets each) while sticking to back squats, deadlifts, overhead press, bench press and power cleans if you want to put on some muscle while maintaining your athletic ability.
reason why these exercises are important are because they're all great Compound lifts. Compound lifts teach your body to work as unit, which is what you do in sports and athletic endeavors (including fighting)
Isolation exercises are something you do to “isolate” a muscle from the system since you don't care At all about athletic performance and only about the size of the muscle. these help work up the “pump” that you'll hear some bodybuilders talk about, amongst other things.
on top of this compound lifts let you get more out of your workouts than one that focuses a lot on isolations (bicep curls, delt raises, calf raises, abductor machine, etc.) so you'll be able to put on a lot more strength with a lot less volume. and most if not all of the size you gain through this style of training will be very functional weight (unlike bodybuilding/sarcoplasmic hypertrophy where most of what you're doing is filling yourself up with fluid for the size)