Beat the game to unlock Zero's mini-game.
Select a game from the Mini-games menu at the title screen.

Complete all missions with 100 points:
Fefnir Mini-game.

Use Zet Zaber for the whole game and beat the game with S rank: Harpuia Mini-game.

Complete every mission with 100 points: Leviathan Mini-game - Use the buster shot for the whole game and beat it with an S rank: Phantom Mini-game.