There are several ways to unlock the Arcade - this is the fastest.
Arcade Secret Fast Unlock
There are several ways to unlock the Arcade - some described in the game, and some hidden.
One of the fastest ways to unlock it is to destroy the boss (warp gate opened), follow the arrow that appears to destroy all remaining enemies, and when the arrow disappears (level cleared), hunt around the level for the below-pictured arcade machine.
It will only appear in a level after it's been totally cleared of enemies. Destroying it immediately unlocks the arcade on the main menu. There are a couple of other ways to unlock the arcade, but this is the fastest :)