Nothing we currently provide will work on CB 1.3. You'd need to develop something to do that or see if you can backport one of our plugins. CB 1.9.1 will work just fine on Joomla 1.5 so there's really no reason to not upgrade CB unless they've loads of custom modifications to core code. If they don't then it's a matter of say 5 minutes to install 1.9.1 and you can use all of our current released plugins. For your particular usage you could use CB Query Field to query their external database and display the data as a field or multiple fields on a users profile, but you'll need 1.9 to use it.

All you need to upgrade is CB it self. This assumes they don't have core modifications in place as upgrading will remove those. You should do a full backup of the site (filesystem and database) encase you need to roll back. Once done review the README included in the CB package for upgrade procedure as I'm unsure if 1.3.1 can be auto upgraded without any manual work.

You don't need to export anything. CB Query Field directly queries the database for the data. It can do so on Internal (Joomla) database or External (provide database connection details). So the database it queries doesn't even have to be on the same server/site.