Here's an easier/quicker way to get this achievement. You can get it right at the beginning of the mission instead of at the very end.
Load the mission Assault on the Control Room
under Halo: CE. Set difficulty to easy.
You start the mission in the back of a Pelican. As soon as you gain control of your character, turn to face the cockpit. Just after the black bars on the screen recede and your HUD fully loads, you'll automatically jump out of the Pelican and you'll see a prompt to hit
(default controls) to get back in. Do so immediately. If you miss it, you'll have to restart the mission.
Now wait in the back of the Pelican while it takes you down the shaft. Switch to Classic graphics mode (
), as it is easier to see when you need to get out of the Pelican. After about 30 seconds, a green fog will start to permeate the cabin of the pelican. Jump out just as it begins to completely obscure your view of the cabin. If timed correctly, you will fall and land on an outcropping from the wall of the shaft. Be careful not to move while you are dismounting or falling, or you may miss the ledge. If you die, you should respawn at a checkpoint just after you reboarded the Pelican.Some people have found the following method of jumping out of the Pelican to be easier. Thanks to Crocsx for the suggestion!
I found it easier to jump back on the Pelican, don't touch the controller at all, and let the Pelican movement move my camera. You just wait, and when you are aiming at the top left angle Pelican, and you start to see the green smoke, you drop.
It's hard to explain without video, but the basic idea is just don't touch your control stick while flying, and when you start to see the green smoke, you jump.
Once you've safely landed on the outcropping, follow it up toward the wall, then turn right and walk along the small ledge that leads to another outcropping. Right in the center of that ledge, you should hear the music cue. Center yourself on the ledge so it continues playing and enjoy the song, accompanied by that familiar, beautiful pop!