To unlock the "Time Trial Challenge" for a cup, you must first beat it as a team, then beat it again solo. The easiest cup to complete would be the Phil cup. I recommend doing this before Hollow Bastion, as enemies gain extra health after that point.
I completed this challenge in the Phil Cup after Agrabah, and was 1-hitting many of the enemies. If you're struggling, Blizzard at the start of the battle is your friend in the Phil cup - there were a few waves I wiped out in 1s by doing this.
Reward for the first completion of each time trial:
Phil Cup: Tech Plus ability
Pegasus Cup: Dark Matter
Hercules Cup: Gravity Break ability
Hades Cup: Save the King (Shield) for Goofy