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

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

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

Unlock New Guns
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