There are a few ways to do this, but here's my strategy.
I used a strategy where you never have to look at the cameras (besides the one with the music box of course). You can stay on that camera the whole night. Simply check the 2 vents, then shine the light at the hall, then rewind. You can spend quite a bit of time rewinding it, since it'll be a little bit before any animatronics appear. If Withered Foxy is in the hall, flash the light into the hall about 7 times, then immediately rewind the box. If any other animatronic appears, whether it be in the hall or vents, drop your mask for about 3 seconds, then quickly rewind the box, even if they're still there (unless it's BB. If it's him, restart the level.) Once it's rewound, quickly drop the mask for another 3 seconds, then see if they're still there. If Foxy has appeared during the time, quickly mash
a few times, then rewind. Drop the camera, then flash him a few more times. Keep doing this, and you should get the achievement. Note that at one point in my play through, the meter just BARELY went under 50%, however I still managed to get the cheevo. I'll eventually post the clip, but for now, follow my strategy and you should unlock it!
Edit: Here's the clip of me finishing the night