In Linux Mint, it should be possible to open tar-archives without installing any further applications.
You can open the archive and put the contents in the data-directory of your wiki; I assume somewhere in /var/www/html
On the plugin-page of backup-tool, restoring a backup is also described in this way:
Restoring a Backup
The following assumes that your Wiki is completely gone - you have nothing left then one backup-archive and the credentials of the Admin-User. It also assumes that you backed up at least your configuration and pages.
1. Start the installation of the latest version of DokuWwiki as if you would start with a new wiki
2. Untar your backup and copy the three directories (conf, data and lib) to your wikis destination-dir, overwriting any possibly existing files
3. You should now be able to log in with your previously used admin account.