As you fall into Bomb Man's room, fall to the right side so you can shoot him on the spot first. Bomb Man jumps away from you whenever you move in & hit him at the same time. He will jump over you a few times before he leaps away from you & chuck bombs where you stand. Evade his bombs then move towards him & shoot so he will do the jumping-over-your-head maneuver again.
Bomb Man is weak against Fire Storm, which you will get when you defeat Fire Man beforehand.
In Dr. Wily Stage 4, you will encounter him again for a rematch. Use the strategy with Fire Storm to defeat him faster.