I doubt it is a virus, that would be very difficult to implement into an ipod and a song added to it would be incapable of causing a virus on it as well because it cannot execute code.
This sounds like a battery issue but it could be a logic issue as well. Replacing the battery in this model iPod should not be that difficult but it will require soldering.
Some helpful steps:
1. Do a full restore on the iPod through iTunes. If that doesn't work we rule out the software being corrupt.
2. Is the iPod hot when charging? This could be a sign of water damage
3. Does the iPod have a hissing sound when charging or when turning on or off? This could indicate a logic board issue.
4. If #2 or #3 are true take it to an apple store genius bar to see what they think.
5. After moving through these steps its now ok to attempt a battery replacement.
Follow the ifixit guide here:
http://www.ifixit.com/Guide/Repair/Installing-iPod-Nano-1st-Generation-Battery-Replacement/774/1
Good Luck!
-Varsity