A good solar battery will not run down overnight with a ARB type fridge, ran mine for four days at Expo. I have dual batteries though. But never got close. The fridge does not draw constinously, only when it is running, which is not that often. Mine draws .5amp normally I think, could be more, never really paid any attention to it. I run a Dometic.
Buy a quality charge controller and you will never have problems. It will tell you how much you are using, how much is left in the battery and the state of the battery. I use a Morningstar digital 15amp in all of my trailers. I have used them solar towers, that I build for the last 12 years. Very good if not the best piece of equipment I have bought for solar.