How you do that depends upon what you use to do the backup. if you make a "clone" of your internal drive then yes that works. Internet connection to re-download Mountain Lion. If you are using a Time Machine backup you would need to boot from the recovery partition and restore files from your backup. That might also mean re-downloading Mountain Lion if you did not include iy in the Time Machine backup.
Here's what I would do. It leaves you with the most flexible option in the event of a drive failure.
1, Download and install
Carbon Copy Cloner.
2. Clone your internal drive to the external drive,
3. Reboot your Mac holding down the option key. When you get to select the drive to boot from select the external drive.
4. If everything is running properly from the external drive go to the utilities folder on the external drive and run Disk Utility to format the internal drive.
5. Now run Carbon Copy Cloner again and clone things from the external drive back to the internal drive.