Nak3d Eli has a great playlist of videos for speed running Halo 2 on Legendary Difficulty. If you're going for the par times you'll probably want to play on easy difficulty (unless Monopolized is also an aim) to make things easier, using the tips and tricks he uses and explains to you with his commentary. The playlist also includes all the Halo 1 campaign missions so for those wanting to skip to Halo 2, jump to the 11th video which is Cairo Station.https://www.youtube.com/playlist?list=PL4RoBZrNqFcoc0lcgUrlG...
Following the tricks and playing on easy will make most of these missions a cakewalk. A few may still cause some difficulty (such as Gravemind and Regret) but if you follow these guides and practice, you'll be able to get through it.
Remember to avoid using any x0 skull modifiers as they void the par times. I'd recommend using the Sputnik skull for Regret as grenade jumps are very important, but apart from that stick to easy, follow the guides and have fun getting this achievement.
All the videos below. Use the link above for the playlist if that's more convenient.