Despite the title and weird box art, this is really just Bomber Boy (part of the Bomberman series). Becomes blatantly obvious once you start up the game and see that the character you control is just Bomberman. Even more so is that game B is literally just a downsized port of NES Bomberman.

Despite what I said, it has a few differences than the usual Bomberman fair in Game A. For one, you can select from a limited number of power ups you start with to power yourself up to have those upgrades from the start of that stage (the usual bomb ups, fire ups, speed ups, etc). Because of that, you can't find upgrades by just randomly bombing blocks. That also means that the game has a shop so you can rebuy most of the upgrades. One other difference it has from the usual Bomberman games is the game lets you pick which area you want to do first. Rather than starting at a fixed point in the game.

Worth a look if you're a fan of the Bomberman series. Especially since this title easily goes under everyone's radar cause it doesn't look like anything anyone is familiar with. Not to mention its on Game Boy. Which unfortunately makes it even more obscure.