Backup your databases. This might including making a copy of the database folders as is and also dumping all your tables to file that are easy to re-import. Very important data should be backed up several times in several forms.
Use your distribution package manager to update the mysql server. Probably something like this: yum upgrade mysql
See if it all worked. If it did, great. If it didn't, hack on it until mysql runs. If your data is alive, great, if not, import it from your backups.