You can do it, especially given that you have a backup (hooray for you! So many folks come here for help, and they do not have a backup). However, you need to take a couple of preliminary steps first:
1. What Mac OS are you running now?
2. From what I can surmise, the latest Mac OS your machine can accommodate is OS 10.13.x, High Sierra. However, assuming your machine has not been upgraded, the internal hard drive spins at only 5400 rpm, and it only has 4 gig of Ram. While all that is "workable", it will be somewhat slow, and at times feel sluggish. If you want to upgrade the internal drive (especially to an SSD, which will definitely improve performance), and the Ram, this link is good:
Even though that is geared towards Sierra, it would be the same for High Sierra.
3. You'll need to insure that whatever third party (ie, non-Apple) software you are using is compatible with the newer Mac OS you'll "get to" (more on that below). That might require downloading and installing upgrades. This link can help you with that:
https://roaringapps.com/apps
4. The process you would go through is the following:
A. Make one more Time Machine backup, just to insure you are current.
B. Start (or re-start) your machine from the (hidden) Recovery HD partition. This link describes how to do that, and what you can do with the software on that partition:
https://support.apple.com/en-us/HT201314
C. You would first select the last option on that screen/menu, Disk Utility. Have it Verify and Repair your Internal drive, and Erase and Format that drive. If Disk Utility says there are issues with your internal drive while trying to do the Repair process, you'll most likely need to replace it. I believe you can have Disk Utility just Verify the drive first. If not, I believe it will offer you the option as to whether or not to proceed with the Repair process.
D. Next, assuming all is well with the internal drive, you would select Reinstall macOS. Note that it says "The Reinstall macOS utility installs different versions of macOS depending on the key combination you used while starting up. For details, see
How to reinstall macOS.". Here is what comes up when one clicks on that "reinstall macOS link:
https://support.apple.com/en-us/HT204904
As you can see, you supposedly have a choice of which Mac OS you would obtain.
In any event, that process will take some time, as the downloading of the applicable Mac OS will depend upon 1) the speed of your internet connection, and 2) the availability of Apple's servers.
E. Once that process is complete, you'll be offered the opportunity to "migrate"/copy needed "stuff" from your backup.
Do you already have n "install MacOS "whatever"" file?
Just to let you know, I would do about this differently.