I forgot to add one other thing. Regarding "you can use an app like SuperDuper to copy the new setup on external drive over the old setup on your Mac's internal drive", what is even better is to 1) Erase and Format the internal drive on your Mac, using Disk Utility on the external drive, 2) do a clean, fresh, "virgin" installation of the OS you have on the external drive, and 3) via Migration Assistant (you'll be offered that once the installation of the OS completes), "migrate"/copy needed "stuff from that external drive to the internal drive. For #2, when you receive Mountain Lion (not sure how it will be via EMail), eventually you'll have a file entitled "Install Mac OS Mountain Lion" "somewhere" on your internal drive. Normally, when you download it from the App Store, it winds up inside your Applications Folder. No matter where it's at, you must make a copy of it first, somewhere else on your Mac. The reason is that, typically, when you launch such a file from your Applications Folder, and complete the installation, that file gets deleted automatically. So, to avoid having to re-download it again, it's best to have a copy of it somewhere else. Once you complete the OS installation onto the external drive, you need to copy that "copied" OS file to somewhere on your external drive.
The recommendation for SuperDuper! is an excellent one, as it is a program that makes an exact backup/bootable clone of what is on the source drive/partition that you are backing up. I have been using it for a number of years, and it has "saved my bacon" a couple of times, for both of my Macs, in the past. There is a similar, excellent program called Carbon Copy Cloner that does the same thing. However, the two differences between those products are 1) cost (SuperDuper! costs $27.95, and Carbon Copy Cloner costs $39.95), and 2) Carbon Copy Cloner also backs up the (hidden) Recovery HD partition whereas SuperDuper! does not. But, for me, that is not an issue, as 1) it can easily be re-created, and 2) when I want to do a complete recovery, I'll boot my Mac to that SuperDuper! backup, use Disk Utility there to Erase and Format my internal SSD, do a fresh, clean, "virgin" installation of the Mac OS that I am using, and then use Migration Assistant to "migrate"/backup needed "stuff" from that SuperDuper! backup to the internal drive. The clean, "virgin" installation of the applicable Mac OS creates that Recovery HD partition. Also, I have a more robust disk maintenance/repair program with TechTool Pro, which does a lot more than Disk Utility, and thus I really do not need the Recovery HD partition (there are other reasons why I do not need it).
Just to "prove" that such a procedure works, while we were on our recent vacation, I mistakenly permanently deleted all the EMails in my In Box (I use Outlook 2016 as my EMail client) on my mid 2013 13" MacBook Air. Fortunately, I had a recent SuperDuper! backup that was only 4 days old, and also I had not received any "critical" EMails in the 4 days since the backup. So, after saving some critical files to my backup, I booted my MacBook Air from that backup, used Disk Utility there to Erase and Format the internal SSD on the machine, did a fresh, clean, "virgin" installation of OS 10.12.3 (the latest version of Sierra, and it was from a "saved" copy of the "Install macOS Sierra V10.12.3" file (I had renamed the file after I made a copy)), and then used Migration Assistant to "migrate"/copy needed stuff from that SuperDuper! backup. I then rebooted my MacBook Air, and I was back in business, with all my Emails restored (I also have to do some additional "upgrades", but as I mentioned above, I first saved some important stuff).