software update
My MacBook air is not showing an option for software update either in-app store or in system preferences.
MacBook Air 13″, macOS 10.13
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The Software Update option in System Preferences will only be there if you are running macOS 10.14 Mojave or later.
If you are running 10.13 High Sierra as suggested by your tag line, then there is no software update option in System Preferences.
If your Mac is compatible with macOS 12 Monterey you can find it in the Mac App Store for download. Or by using the link here: link-> How to upgrade to macOS Monterey - Apple Support
If your Mac cannot upgrade to Monterey, but can support a slightly older macOS version but still newer than High Sierra, then you can get a compatible version from here: link-> How to get old versions of macOS - Apple Support