Hi Markus,
to be on the save side you should keep the same hostname. It's possible to change it, but it isn't that simple. If you want to change it have a look at those 3 articles:
ZmSetServerName - Zimbra :: Wiki Logical hostname - Zimbra :: Wiki SSL Certificate Problems - Zimbra :: Wiki
But if you keep hostname/ip as it is, you shouldn't have those problems. The reason why I recommend to uninstall Zimbra before upgrading Ubuntu is that I'm not sure how the Zimbra packages will behave while upgrading the distribution. Those packages are not part of the Ubuntu repositories so they won't get upgraded automatically. This means you have to reinstall the packages for Ubuntu 8.04 anyways.
It might just as well work without uninstalling Zimbra first, but just to be save I would invest those extra 2 minutes.
But again, I think it's really important to first upgrade Zimbra and really test if everything is working fine (check logs etc.) and then do a backup. Otherwise it's really difficult to see where possible errors are coming from. Also you have the possibility to reinstall Ubuntu 6.06 and use your backup again.
Ah and another thing, when copying /opt/zimbra make sure you preserve owners and permissions.
Again good luck
Conny