After searching around on Google, here's what I've got:
You'll need to use this one Fish Hook item for the battle. To get that, you need to use a blue door on a ledge south of the shop in World 3. Once you have the Fish Hook, when you're in battle with the Cheep Cheep you use the hook when he jumps into the water. Doing so will fish him out and then you attack him as much as you can. If you aren't quick enough, he'll explode and deal massive damage (99 damage to be exact).
Hope this helps!