I had that problem once and all I did was reset it. Hold down Menu and Center for 3-6 seconds and it will reset. It should work normally after that.
When an iPod Nano turns on it should show the Apple on the screen right? No matter how long it has been off for?
That's not the case. It only shows the Apple when it has gone into deep sleep mode [I think after 24 hours of inactivity].