This achievement is exactly the same as the 360 anniversary achievement "How Pedestrian."
In order to get this achievement you will have to walk everywhere in the mission which adds some time on the clock but other than that it's pretty simple still.
The only part that you will have trouble with is the gap you're suppose to use the warthog to jump but obviously can't since it will void the achievement. Instead of using the vehicle at that part you can perform a grenade jump to get across. In order to perform one you will need a grenade to toss at the ground then jump in the direction you wish to be propelled when it explodes to give you an extra boost.
Doing this with a friend is easier as you can run ahead and wait for a checkpoint to be given right at the gap. When you get one you can try the grenade jump and if you fail then just reload the checkpoint. when you make the (Grenade) jump over the gap then just run ahead and your friend will teleport to you.
second CO-OP method is to have a friend drive the warthog and you not get in it at all and you will teleport to them after they jumped the gap in the tunnel.
Here is a video by Rooster Teeth showing how to get the same achievement on the 360 version of Combat Evolved