NFS deals with permissions the same as for local filesystems. So as long as the user on your new installation has the same UID as the owners of the files you're trying to get at, then yes, they can access the files. Check your /etc/passwd file and use the same UID's and you'll be fine (but I would caution you to use some UID below 1000 but above 100 for your backup user, so that it won't conflict with default system users installed on most systems. Backup is one such user, but it's a different UID and GID in most distributions, so don't depend on it!)
Remember that the user NAMES don't really count. They're just cosmetic. You could have two users "joe" and "job", and they could both have UID 5600, and they'd be the same user as far as the system is concerned. So creating a "backup" user on your new CentOS system doesn't necessarily mean it will have access to the files. It MUST have the same UID.
Good luck to you!