Ipod shuts down by itself
My ipod touch 1st gen shuts down by itself even while it has 50% battery still on it. However, interestingly while it is being played or charged, its stays on. I have already done restore and the problem persists
iPod touch
If you successfully restored (went through all steps without error) to factory settings/new iPod via iTunes (Not via Settings>General>Reset>Erase all Content and Setting) and still have the problem that indicates a hardware problem.
Make an appointment at the Genius Bar of an Apple store. However, Apple may not be of much help with your old device. I suspect a bad battery especially if it is the original battery
Apple Retail Store - Genius Bar
http://www.apple.com/retail/geniusbar/
If you are handy you can try replacing the battery yourself. A new battery costs less than $15. See: