If you don't run the backup on a server but on a client you can activate the desktop notification for a user that is logged in. It uses *notify-send* to show notification messages.
In a cronjob that start the backup script as root must know the user that should get the notifications.
Set a variable `SUPER_USER=<USERNAME>` in your cron file.
Set a variable `SUDO_USER=<USERNAME>` in your cron file.