One of the more frustrating achievements in the game. If you are struggling to unlock this the normal, and constantly getting a very close time, but not quite there, you're in luck.
Firstly, to unlock the mini-game complete the Hobbe Caves level, and choose the right hand side path.
Now you have 1 of 3 options:
1) Do the achievement normally. (can be frustrating)
2) If you have a turbo controller, you can try using it for the achievement. Certain brands will do the trick, but some won't.
3) If you have a friend who has unlocked this mini-game, have them start an online session and invite you. Have them set it up so that the characters are in inside the carts, and waiting for him to choose "I Understand!", right before the race. Now join his game (using his invite) and have him start the race anywhere between 7-22 seconds (depending on connection speeds) after you accept the invite. You will join DURING the race, and will be added halfway through. Your timer will start from when you join allowing you to easily register times under 13 seconds. If your friend starts too early, you will join after the race ends. If your friend starts too late, you will join before the race begins. This may take a few tries for you to nail down the timing, but this is the easiest way if you are struggling. Just adjust the time your friend waits to start the race accordingly.