or via updating the file via ftp and then reinstall magento again on the same db
But even this will break the system completely (be prepared to fix bugs) and try everything on a staging environment first.
this will keep you live store intact
So theres no way I can backup the necessary info on the current server and install fresh 1.6?
you can make a backup in case of any accidents. And then migrate your shop to fresh Magento installation. I did it with help of cart2cart http://www.shopping-cart-migration.com Quite fast and effective. Not free but the price is affordable.