i need to update my MacBook air

The model year of a MacBook Air is available at Apple menu > About This Mac.

A mid 2009 MacBook Air can be upgraded to El Capitan 10.11.6 but no later version. See: How to get old versions of macOS - Apple Support

A late 2010 through mid 2011 MacBook Air can be upgraded to High Sierra 10.13.6 but no later version. See: How to get old versions of macOS - Apple Support

A mid 2012 MacBook Air can be upgraded to Catalina 10.15.7 but no later. See: How to get old versions of macOS - Apple Support

A mid 2013 and early 2014 can be upgraded to the latest version of Big Sur macOS 11 but no later. See: How to get old versions of macOS - Apple Support

An early 2015 or later can be upgraded to the current version of Monterey macOS 12. See: How to upgrade to macOS Monterey - Apple Support

..a74phillips:

Perhaps if you utilize the information on MacBook Air model history,

and identify your Mac by build date, and year.. That could help, since

there are only ..so many dozens of MacBook Air models.. otherwise.

• Identify your MacBook Air model - Apple Support

//support.apple.com/en-us/HT201862

Or look into this page, where 'downloadable-installers' are available..

to see & identify which one(s) may work for your older MacBook Air.

• How to get old versions of macOS - Apple Support

//support.apple.com/en-us/HT211683

You may wish to avoid the newest macOS, tho some says 'supported.'

Sometimes those need an in-between system, to ready the Mac; for the

newest one. That may mean, in order to get/run Big Sur, you may need

to first get High Sierra (full systems/download). Then try that newer one.

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.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence.

If no ” Get Button “  Replace the https in the page's address to macappstore. Credit to Contributor @Niel

Computer Qualify for the Monterey upgrade

macOS Big Sur 11 hardware requirements

macOS Catalina 10.15 hardware requirements

macOS Mojave 10.14 hardware requirements

macOS High Sierra 10.13 hardware requirements

Caution - jumping over all the newer versions and directly to Big Sur  / Monterey on Qualifying computers is Technically possible - there are those who have attempted and were successful. Then there are those who attempted and had less than good results.

If this is a Production or Mission Critical Set-up - a more prudent and measure approach is suggested. 

The above is from observations on the Apple Support Community Forums and IMHO

Tips Big Sir Upgrade Released Nov 2020

Tips Monterey Upgrade  Released Oct 25, 2021

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.