Easy mode:
Beat Wario.

Thrilling mode:
Beat the game in Easy mode with a high score.

Hard mode:
Get at least 15 mails in Thrilling mode.

Boss mode:
Beat the game.

Staff mode:
Beat the game.

Chiritorie mini-game:
Beat anyone in the first group of characters.

Chicken Race mini-game:
Beat any two of the first group of characters.

Dong Dong mini-game:
Beat anyone in the second group of characters.

Hurdle mini-game:
Beat any two of the second group of characters.

Dr. Wario mini-game:
Get 15 mails against Blue Jimmy.

Fly Swatter mini-game:
Get 20 mails against Yellow Jimmy.

Paper Plane mini-game:
Beat Red Jimmy.

Wario's Sheriff mini-game:
Get 25 mails against Red Jimmy.

Pyoro mini-game:
Beat the Wario stage.

Pyoro 2 mini-game:
Beat the challenge score on every game.

Thanks to Revolution readers Harvey Wallbanger and rufus33!