Here are two shortcuts you can use in combination to get a better time on the track. First, drive around until you are near the section of track with the "U" shaped turn. Run off the track and into the water. Lakitu will tow you to the gap. From here, go across as if you were going to hit the other side of the gap, but veer off toward the left. Lakitu will then to you to the finish line.
Mario Kart 64 Shortcut Video Walkthrough Series for Nintendo 64
Mario Kart 64 - Royal Raceway Shortcut Video
