Yes you can authorise more than one computer to the same account (I think the limit is 5 computers) but that won't move the music that you've bought with that account.
Which in a way rolls on to this...
"Can I just sync all the music onto the ipod (it will just about fit on) and then simply plug the ipod into the new computer - having installed itunes of course?"
No. And if you plug that iPod full of your music into your empty iTunes at home then if you just select the default options (automatically sync/manage the iPod) it's going to loose everything.
The reason for this is that iTunes tries to make things very easy for you. So you plug an automatically managed iPod into iTunes and it sees that there's an album on there. It also sees that that album doesn't exist in your iTunes library. "Ah-ha" thinks iTunes, you must have deleted that album from iTunes, and you've told me to automatically sync that iPod, so I'm going to automatically delete it from the iPod too.
In short if music is on your iPod and not in iTunes then you need to manually manage the iPod. Instructions for this are here:
support.apple.com/kb/ht1535
But you can move music from one iTunes to the other, so if you have all your music in iTunes then nothing will get deleted from the iPod.
To do this you need to get the music from your work PC onto your own PC, and this is pretty simply, especially if you've got an external disk drive. Simply fire up up iTunes on your work computer and drag the music off onto the drive, then plug the same drive into your home computer and drag the music from the drive into iTunes.
If you don't have an external hard disk drive then there are other ways, but using one is by far the easiest.