How long can a frog that is not even 1 inch long survive without food?

What kind of frog? I think that a frog can go a few days without any major issues, normally. But, if this is a tree frog, then less than an inch is pretty young. He might not be eating because he's really stressed. Tree frogs shouldn't be shipped to pet stores until they are at least an inch long because the moving (into a container, then the shipping, into a new environment, then shipping to a new owner and yet another new environment) is just too stressful for them. If you think this is the case, then all you can do is add some really tiny crickets, make sure he has clean water and enough heat (if he's a red-eyed) then force yourself to leave him alone, so he can get relaxed and comfy.