obsolete MacBook Air...can I update?

Hi,

That Mac can be upgraded to 10.15.7/Catalina: How to get old versions of macOS - Apple Support 

Before upgrading, backup, check apps and peripherals for compatibility with Catalina.

Catalina will not run 32 bit apps, so download Go64 and scan your Mac to see which of your apps are 32 bit and will need to be upgraded/replaced: https://www.stclairsoft.com/Go64/

Some apps may be 64 bit but still not run on Catalina; you can check them out here: https://roaringapps.com/apps