Unlocking Time Attacks in Shadow of The Colossus Shadow of the Colossus Guide, walkthrough

Time Attacks become available only after you've managed to complete the main storyline (i.e. defeating all 16 colossi). Once you've managed to complete the game, you'll receive access to New Game+ mode that can be activated by loading your saved game. Unfortunately, the difficulty level of New Game+ will be the same as the difficulty of the chosen save game (easy/normal/hard), which means that you only receive access to the challenges from the given difficulty level. Having this in mind, it should be mentioned that the time attack rewards require completing normal/hard challenges.

You can challenge every colossus you've encountered during the game

You can repeat the colossus fights at will or restart the challenge during its course. To begin a Time Attack, approach the sculpture of the chosen colossus and press (R2).