You need to use a light or medium fighter for this. A bomber can't complete the tracks because you need to execute a loop for the high and low rings. And time is limited, so you need to hurry. I recommend a medium fighter because you'll need to make some very precise maneuvers and you risk overshooting in the Arrow or Darket. I completed the game in a Dralthi Rhino.Don't forget to take a cloak if you'll be doing this in a public game. If all else fails, drop by #Wingnut for some help setting up a quiet bearpit room to practice in.
The right stick is instrumental here. Roll left and right to quickly make your way down the track. Keep the right stick pushed back to catch a low ring. Keep it pushed forward to catch a high ring. Keep one eye on the counter at the bottom of the screen. You'll see a number flash when you succesfully move through a ring (this is not always obvious for the low and high rings depending on your camera angle). For some parts you may need to back up a little and realign your ship or you won't be able to do the looping. Be careful that you don't back up through the previous ring or you'll have to start over! Save time by using afterburners for straight sections.
You need to complete the ring game three times in the same bearpit game (it's okay to die between turns). Each time you will be presented with a harder track. You should set up a private game with three other players. Two players can play the ring game (the two ring powerups are located at opposite sides of the arena) while the other two hang out in the pit.