Stage Unlock: Yoshi's Island Past Stage:

To unlock the original Yoshi's Island stage from the N64 version of the game, use Yoshi and hit over 1,350 ft. (400m in the JP version) in the Home-Run Contest. To do such, stand next to the Sandbag and TAP the X or Y buttons to jump, then use the Down + A move to hit the bag for around 30% damage each time. Do that about 4 times to get the bag up to around 100-110% damage. Since you won't have much time left, get the bat and use a Smash hit to send the bag flying. If you did it all well enough, you should make it well over the required distance.