This has a lot to do with how smart your robot — or those who stored the robot — are.
the Dead Sea Scrolls, made of leather and papyrus, survived nearly 2,000 years in clay jars because (basically) they were in a dry, non-reactive location. The jars kept the critters out and the desert kept the humidity out. But there was decay...
... But you're dealing with an automaton. Your technology is better. Your circumstances can be better.
My point is, how you've packaged the robot for storage determines whether or not your robot will stand the test of time. Are you using something like magnetic floppy or hard drives? They won't last. Solid state? That might last. Optical? It might last (the plastic used in optical disks would need to be stored very flat or gravity over that period of time might deform them).
This, of course, assumes that we have no better tech to deal with than what we have today — which is unlikely since we don't have independent automatons today. So I'm voting that if the robot could intrinsically last 1,000 years in normal operation, it'll last 1,000 years in storage.
So, pack him in an hermetically-sealed box with humidity control and you can story him beneath Niagra Falls for a millennium. Or wrap him in plastic wrap in a cave in the Sonoran Desert. Either one would be believable.
However! If your robot can't last 1,000 years in normal operation, then what is it about the robot that will need to be replaced to make it operational again? The power source? Motors/pneumatics/hydraulics? A lube job? One basic rule of engineering is the more complex the design the more complex the maintenance (generally speaking). Your story can always declare the robot's survival to be so and who would we be to say you're wrong? But if you want something with a taste of realism, you'll need whomever finds the robot to fix something.