Sorry, I didn't even see you'd mentioned fedora.
You'll have a slightly more tortuous route with FC.
With the need to change operating systems and upgrade to a current release the simplest thing would be to (install new new server & Zimbra) use imapsync to transfer the mail, export the contacts as .csv file and calendar information via rest then import them into the new server. There is also a wiki article on
Migrating from OSS to NE which should give you some food for thought.