The cactuar ghost is the ghost you get which shows your best performance in the speed run.

Here's the tough part: You must set a record of under 00'54''000 on the Beginner Course. If you succeed, start Time Attack mode and pause the game. Go to options and then ghost. You'll notice a new "? ? ?" option at the bottom. Select it and a running cactuar will replace the moogle. The ghost and your record can be saved on your file.
As I said, the ??? is intentional. The menu is in english even in the japanese game.