I am here to say you can because I did it with my 1st Gen iPod.
The key I found was that when you plug it into the Windows machine, format the iPod drive like you would format a hard drive in Windows. Then run the iPod updater for Windows and you should be good to go.
ur pc must have firewire for ur ipod to work... i don't think the first gen ipods supported usb 2.0.. i think that came out later.. i could be wrong... what id do is just dl the ipod software updater and run that... u can get it here..
if all ur songs are on that ipod.. make sure u back em up with somethin like ipod rip or some other ripping program... then id hit RESTORE on the updater to make sure everything gets cleaned out on that ipod... once thats done... u 'pod should work with ur pc and itunes just fine