On the main menu, press: X thirteen times, SQUARE, X five times, SQUARE, X two times, SQUARE, X nine times, SQUARE, X seven times, SQUARE, X three times, SQUARE, X eight times, SQUARE, X five times, SQUARE, X, SQUARE, and then X twenty times.

To unlock new guns, beat the game in Arcade Mode to unlock the new weapon. Here's how:

Beat with pistol: machine gun
Beat with machine gun: cane gun
Beat with cane gun: shotgun

To unlock Hyper Arcade mode, beat Arcade mode with the shotgun.

Beat all the Mighty levels in Cadet Training to unlock Mighty Joe Jupiter Trilogy (in MJJ mode).

Finish all episodes of the Mighty Joe Jupiter Trilogy to unlock Music Juke Box, which can be found in the options screen.