When you go to Sector 5, you have a better bomb which isn't timed. Good thing too because you've gotta do that stupid "press all 3 buttons at once" thing on the way out.
Beating Airbuster (I believe that's its name) in one hit isn't hard to do. A blast from behind of Barret's Big Shot should be more than enough if you've got him leveled up enough.
The game may seem easier because you're older now and probably far more experienced as a gamer.