I need to update my Macbook air and it is not showing the 12.13 macOS update

To be clear, software update generally means to install software to update the macOS currently running on the Mac; update Catalina 10.15.1 to Catalina 10.15.7, for example.

To update your current macOS, click the Apple  icon > About This Mac and then click Software Updates.... If there are any updates available for your Mac, they will be listed and you can apply them when you like.

A software upgrade is usually a generational step up to a newer macOS; upgrade from 10.13 High Sierra to 10.14 Mojave, for example, or from 10.15 Catalina to macOS 11 Big Sur.

❗️ Do NOT upgrade until you have made a Time Machine or other backup of your startup drive.

Learn How to Back up your Mac with Time Machine

How to Upgrade  macOS:

The macOS version you can upgrade to depends on the year your Mac was released.

Click the  (Apple) menu at the top of your screen and choose About This Mac.

Check the Compatibility of your Mac with newer versions of macOS:

➡️ macOS 10.13 High Sierra is compatible with the Macs listed here

➡️ macOS 10.14 Mojave is compatible with the Macs listed here

➡️ macOS 10.15 Catalina is compatible with the Macs listed here

➡️ macOS 11 Big Sur is compatible with the Macs listed here

➡️ macOS 12 Monterey is compatible with the Macs listed here

❗️ – IMPORTANT – Your 32-bit apps won’t work in Catalina, Big Sur or Monterey.

https://support.apple.com/en-us/HT208436

Use the free utility Go64 to check the compatibility of your apps:

https://www.stclairsoft.com/Go64/

✅ - IF your Mac and apps are compatible, then using Safari only, download the macOS upgrade installer app:

Click > High Sierra,  Mojave,  Catalina,  Big Sur  or  Monterey.

Once downloaded, double-click the “Install macOS [version]” app in the Applications folder to begin the upgrade.