That's an easy one: No.
The sync software in Vista doesn't support WinCE 3.0 devices like the Jornada 720. Sorry.
As for the battery, the original battery will probably be good for only a few hours. But you can find a replacement on Ebay. I bought mine from the following seller, and it's good for around 10 hours.
http://myworld.ebay.com/enessystore