Are you planning to do an in-place upgrade of the OS (ie stay on the same server)? Personally, I wouldn't recommend this. Not because of any bad experiences I've had; just in general I like fresh OS installs on servers.
As far as the Zimbra upgrade from you shouldn't really have any issues there. I recently did an upgrade from version 5 to 6 and it took about an hour (I skipped the database check). Obviously make sure you have a good backup prior to attempting the upgrade. Check the release notes on version 6 -- there may be a recommended intermediate upgrade on the version 5 side of things.
If you are planning on moving to a new server you must install the same version of Zimbra on your new server as you are running on your current server. So, in your scenario you would either upgrade your existing Zimbra install to version 6 (6.0.5 or whatever the latest is when you actually upgrade) and then migrate to the new server running CentOS 5. Or migrate to the new CentOS 5 server and install the same version of Zimbra 5 you are currently running. Then upgrade to Zimbra 6 after you are up and running. Check the wiki and forums for info on this type of migration/upgrade.
Also, just want to point out something since you say you inherited the admin of your Zimbra server. Your profile shows you are running the Network version of Zimbra. You do realize that this is not an "officially" supported OS, right? (Although a lot users feel it should be --
bug 23487)