[look 171]

No sander. You will end up with valleys and hills and you will want to throw both sander and yourself out the window after a couple hours covered in dust.

Someone made those cabinets, not bought from a bog box store.
Lets split hair for a second. That side looks to be an imported plywood so its more then likely a little thinner then 3/4". You need 11/16, so that's only 1/16 you need to shave down. Technically that will work, but somehow, the gods ain't with people like us on such horrible product, plywood, so you are correct, 1/8 is what you need ( doesn't matter thickness of plywood at this point). I know you have a router, but if you really want to abuse yourself, make a carriage along with a 3/4" straight cutting bit and set the router to cut 1/8 off after 1000 passes to achieve the 1/8" off the side of the cabinet. Its actually more accurate then the power plane. Me, I buy the plane and have everything in place, be done before dinner.