Stuck with High Sierra - No Updates?

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install. Make sure to use Only Safari to commence the download as Others may not work

Make a Time Machine Backup Before attempting. Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina and Big Sur Only support 64 bit and there is no workaround.

Are there any AntiVirus, Disk Cleaner, Optimizers, Defrag, VPNs etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal operation of the OS and may even Corrupt the OS requiring a Reinstallation.

ClaireFlaubert wrote:

So basically once he frees up some space the download might become available instead of the “no updates” thing?

Thank you !!!

no, the upgrade through software update would only be available if the mac was on catalina. to jump from high sierra, just click on the link i provided in my 1st response. it will take you there directly where you will be able to "get" it.

i should have added that any apps on high sierra that are 32bit will not function at all on big sur. apple dropped support for them starting with catalina. they will need to be upgraded or replaced with comparable 64bit apps.

the 2018 MBA definitely can be upgraded to big sur, but 7.5GB of free space is nowhere close to enough free space to perform the upgrade. to upgrade to big sur from high sierra you officially need 35.5GB free. and it's even been recommended in this forum to have 50GB free. once you have made that much space available, click on Get macOS Big Sur and that will take you to the download in the app store.

Follow the advise of @ jeffreythefrog.

Further reading for the eventual upgrade to Big Sur and Before attempting >> Tips How to Upgrade to Big Sur  <<<