FYI, an update is a patch to the current OS. You are looking for an upgrade to a new OS.
Try searching the App Store for "macOS" to see if it locates the OS installer. Apple used to have it front and center, but has taken to hiding the upgrade option. Sometimes you can see a link to the macOS upgrade on the right hand side of the App Store. I'm not sure, but I would think your laptop should be able to run High Sierra.
You can also try downloading the High Sierra OSX 10.13 installer from here. You might want to create a bootable macOS USB installer just in case something goes wrong so you can make another attempt at installing or repair. You should test it to make sure it boots.
Also make sure you have good verified working backups before you begin. It is much easier to restore data from backups than trying to recover your files from a non-booting system if something goes wrong.