So the upgrade process is pretty straight forward and will not effect your Exchange server. That is unless you have some sort of custom security scheme that will prevent Exchange from working with normal system accounts.
Then I would re-examine the 4gb partition and try to upgrade that drive to at least 8gb to be on the safe side (not necessary but you will thank me later)
Is the c drive on raid or a mirror? That would make this easier. Otherwise partition magic or ghost should help you out in enlarging the drive. (You did not say how big the drives are or their setup)
Once the hard drive is bigger put the windows server 2003 cd in the running windows 2000 server and follow the prompts for an upgrade.
Are you using active directory? This adds some more steps but the wizard should be able to walk you through that.
Long story short if you have a default installation – update your machine (Service pack, windows updates the whole 9) then run the compatibility wizard followed by running the installation. It is so worth it just for the shadow copy features?
What does this server do? If it is able to be formated that would be optimal if not the above mentioned upgrade process should do you just fine.
Any other questions or concerns, please update this ticket and I will respond! Good luck and welcome to 2003!