If you restore the database then you shouldn’t need to import anything from uSync. It could be that the uSync files are out of date - I would take the database as the source of truth, as that was running on the existing server.
Was there content in the site on the existing server? If so, then a backup and restore should be adequate. You may want to double check the existing agency have given you a backup of the correct database and that you didn’t create a new database by unintentionally, either when setting up the instance or running Umbraco for the first time.
I don’t know what your server control panel looks like, but if you have the facility to restore the database again I would try that first.