How can I update my (super old) Mac OS when the App Store shows no available updates?
I would start first opening this link in safari.

https://support.apple.com/en-us/HT208969

Don’t take the bait in step 4 trying to get you Catalina, You want this next link first...

https://itunes.apple.com/us/app/macos-high-sierra/id1246284741?ls=1&amp;mt=12

While it’s downloading check your backup is solid. The upgrade and updates don’t intentionally erase data, but better safe than sorry.
Also, seriously consider an internal SSD if you move to APFS and past High Sierra.