This stage is not terribly troubling to complete without the use of brakes. Keep your speed down the best you can manage and make use of the (RT) to quickly adjust for excess momentum. Also utilize the track's railing and curbs to direct your ball/marble properly. You are given a mandatory checkpoint as well as a manual checkpoint. Plan the use of your manual checkpoint wisely.
If you are having difficulties completing the track in this normal fashion, try the method shown in the above game clip. Follow the path past the two hard right angle turns and stop halfway through the first rounded curve. Make a checkpoint. Turn your camera to face the exit and then attempt a jump to finish off this level. There are guard railings along this final stretch of track that can make the jump a little frustrating, but all together not too challenging.
A perfect run can easily yield a sub-ten second completion time. Do not use your brakes (LT) at any time.
Good luck and happy achievement hunting!