Puck's mini-game:

Successfully complete the game on the easy difficulty setting.

Full pause screen:

Pause the game and press X + Y.

Battle arena mode:

Successfully complete the game on the normal difficulty setting.

No limit mode:

Successfully complete the game on the hard difficulty setting.

Easy hard mode prizes:

Use the following trick when obtaining the hard mode prizes. Set the difficulty to easy and proceed to the final stage save point (before fighting Balzac's first form). Quit and save the game. Set the difficulty back to hard and continue the game. Even though you completed most of the game on the easy difficulty setting, you will still get the prizes for hard mode. You still have to complete the final stage in hard mode, but some tough sections (such as the battle with Zod) are a lot easier. This can also be done with the normal difficulty setting if needed