Get to full speed (preferably but not necessary) and press 'N' for neutral. Now you can cruise along at full speed with easy steering and invulnerability. As long as you stay in neutral you will have unlimited time as well.

At the start, turn the car around 180 degrees. The display should now show the WRONG DIRECTION message. Head off under the bridge until you get to a split in the road. Take the right hand branch and keep going until you reach a checkpoint. Go through and turn the car through 180 degrees until your heading the other way. Go back through the checkpoint in the right direction and you should hear a ping. Keep going to the finish and once under the bridge let the clock run down to zero. You should now get a message saying that you've qualified for the race!!