EDIT: xTGE mentioned that this can be easily achieved with the bandana skull activated, which gives you infinite ammo.
This achievement is quite easy to get. I just got it on Heroic with little difficulty and 47 shots left, so it should be a cake walk on Easy. As far as I saw, I didn't see any Marines carrying any snipers, so your best bet is to stick with the one you start with.
The sniper rifle is most useful at the beginning of the mission, when working your way toward the Covenant ship. Once on the ship, the sniper becomes basically useless. I would recommend replacing your AR with the Plasma Pistol as soon as possible since it is very effective at taking out Grunts and Jackals quickly, and for taking out Elite's shields.
When fighting the Hunters (there are at least 6-8 in the level), if you constantly run around them, you can melee them in the back without them being able to shoot or hit you. This method can potentially save you health and ammo and is very easy to learn.
Make sure you constantly have the sniper on hand and don't waste too much ammo, otherwise, restart the checkpoint. Using the Plasma Pistol and being on a lower difficulty is more than enough to get through the mission with well over 4 shots left.
If you vote negative, please tell me why in the comments so I can try to make this solution as straightforward and helpful as possible.