If you have a saved game file from One Piece Unlimited Cruise: Episode 1, you can transfer boss characters and your character stats.

To unlock Boss Rush mode, defeat the One Piece Unlimited Cruise Bosses with the Fury Boost and collect the map pieces on Unlimited difficulty.

To unlock Survival mode, beat the game on Normal difficulty.

Unlock the following characters in Vs. mode by defeating them on the corresponding difficulties:

CHARACTER - HOW TO UNLOCK
Crocodile - Defeat him on the Normal difficulty.
Doflamingo - Defeat him on the Normal difficulty.
Edward Newgate - Defeat him on the Normal difficulty.
Final Boss - Defeat him on the Normal difficulty.
Final Boss (alternate form) - Defeat him on the Hard difficulty.
Garp - Defeat him on the Unlimited difficulty
Human Lucci - Defeat him on the Normal difficulty.
Kaku - Defeat him on the Normal difficulty.
Kid - Defeat him on the Normal difficulty.
Kizaru - Defeat him on the Normal difficulty.
Leopard Lucci - Defeat him on the Normal difficulty.
Mihawk - Defeat him on the Normal difficulty.
Pauly - Defeat him on the Normal difficulty.
Shanks - Defeat him on the Normal difficulty.
Smoker - Defeat him on the Normal difficulty.
Vivi - Defeat him on the Normal difficulty.