Joe and Hand are right - if that back is 1/4-in you'll need to support it better (and ignore my post, since I was picturing a 3/4 back like I had).
A good way to do that, using scrap plywood, is to put vertical strips on each side and, perhaps, another horizontal strip across the middle. That way, you backing gets supported all the way around. I don't know how you'll address the junction of the backing to the box sides.
The vertical "cleats" go on backside of back, inside the sides, between the two horizontal cleats, same thickness as cleats. Fasten from outside cabinet thru the sides, then fasten back to those verticals. That will keep back from bowing, especially if there are shelves, with back tacked to them. If there is gap between back and sides inside cabinet, cover with 1/2"X1/2" moulding ( square or quarter round or ?) fastened thru back to cleats, ( requires a notch in back corners of shelving) or rip narrow strips of same material as back and fasten them flat against frontside of back. If you can live with narrow gap behind shelf, no notch needed. Possibly you could run caulk in inside corners to cover any gap after putting verticals on back.
I dunno if I cleared up my instructions or just made them muddier.
Measure twice, cut once.
Look at the nail, not the hammer. Watch the fence, not the blade.
If you hook your thumb over your belt you won't hit it with the hammer or leave it layin on the saw table.