Can I restore a system upgrade from a Time Machine backup?
Unless you excluded the OS files from the backup, Time Machine should've backed that up along with everything else; but in order to recover the entire OS (rather than just your user files), you have to do the restore differently: use the restore tool in Recovery Mode (reachable by starting the Mac with Command and R held down). However, there are a couple of warnings I should give about this route:


I haven't tested it, but I think this form of restore will wipe out everything currently on the disk, replacing it with the earlier backup snapshot. Have you done anything important since the backup? If so, you should back that up before going this route.
If you'd excluded any files/folders from the backup, they'll be missing from the restore as well. It's fairly common to exclude the OS itself from backups, so if you'd done that this form of restore will not give you a bootable system.


Because of the above, I'd be inclined to stick with the restore you have, and simply reinstall the update.

BTW, the question seems to contain a bit of a misunderstanding about how OS updates and backups work. Time Machine (and other backup systems) back up files and folders. An OS update isn't a file or folder that can be backed up as an item; instead, it's a bunch of changes to various system files and folders. If those files and folders are being backed up, the updated versions will be backed up; if not, they won't be. Similarly, you can't restore a system update, you can only restore the changed versions of the various system files (and since you don't know precisely which ones were changed, that pretty much means restoring all of the system files and folders).