Here's how you get the cat:
1. Find the Cat: The cat is located in the Temple of Time just before you travel to the future. It's hiding in the back of the temple, behind the entrance to the Gerudo Training Ground.
2. Use the Ocarina: Play the "Song of Healing" on the Ocarina to lure the cat out. The cat will follow you to the back of the temple where a fairy is waiting.
3. Get the Fairy: The cat will lead you to the fairy who will then fly away with the cat.
The cat is just a small detail in the game, not directly related to the main storyline. You don't get any rewards for finding it. It's more of a fun little side quest that adds some charm to the game.