EDIT: December 9, 2021: It has come to my attention that MCC no longer allows players to select prologue/epilogue missions from the mission select screen while in co-op, therefore I am updating the guide to have valid information.
However, I have found out that you can still access the prologue missions and get the achievements in co-op by using the full campaign playlist.
So, the fastest way to get this achievement now is to put in a second controller or invite a friend to your lobby, go to Campaigns, select Halo 3, then Playlists, and scroll down to select the “Legendary Campaign” playlist. After starting the playlist, either watch or skip the opening cutscene and after it is over, the next mission will load and you will unlock the achievement.
This will also unlock two other co-op specific achievements, "Standard Operating Brocedure" and "Brovershield," if you haven’t unlocked them already.