I think you would set up your new server and then restore your database backup.
After that you would then apply the database update scripts one at a time which is explained here.http://www.asterisk2billing.org/documen ... a2billing/
Looking at the explanation you might want to alter the database update script so that it only updates up to the version of your current A2billing database when you first install it. Then after you restore your database backup of the same version...run the rest of the updates to the current version.
If it goes pear shaped for some reason you still have your existing server in production before switching over. You can just delete the database on the new server and start again until you get the right recipe.