You are pitted against a Giant Yoshi. All you have to do is simply kill Yoshi. Since Yoshi is big, it will take a lot of damage to kill him. Giant characters have a disadvantag though. First of all they are a big target and are easy to hit. They're jumping distance is reduced and they fall faster. Use smash attacks and items to knock Yoshi off the stage. Often times when you fight near the edge, Yoshi will kill himself while trying to attack you.
Use Jigglypuff. When Yoshi comes toward you, use rest. wince he's so huge, you most likely wwill hit him, and he'll just drop off the edge. you can get records less than 00:00:52 this way.