Trying to update a MacBook Air mid 2013 from macOS Mojave to macOS Big Sur

For correctness purposes only.

An update is from within the same version of macOS. As in Mojave 10.14.5 to Mojave 10.14.6

An upGrade is moving from Mojave to a newer version of macOS this computer Qualifies to run.

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.

macOS Big Sur 11 hardware requirements

Empty Space Requirements specific to the Big Sur. Apple states 35.6 GB Empty Space but suggest 50 GB

macOS Catalina 10.15 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

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.

Try while in Safe Mode.

How to use safe mode on your Mac - Apple Support