Is it necessary to update macOS when WhatsApp Web prompts to update macOS on MacBook Air 13"?

What is the hardware model year of your 13" MacBook Air? That will determine whether, and how far, you can upgrade macOS.

If it is

• Before 2015 – You cannot upgrade past Big Sur
• 2015 or 2017 – You can upgrade as far as Monterey
• 2018 or 2019 – You can upgrade as far as Sonoma
• 2020 – You can upgrade all the way to Sequoia

How to download and install macOS - Apple Support

If this is a Web site, you could also try installing Firefox and accessing the site through Firefox. The current version of Firefox will run on Catalina and higher, so you could run it even on Big Sur.

Get calling, screen sharing and faster experience with the Mac app. Requires macOS 12.1 or newer. 

macOS 11 Big Sur was released in 2020 making it a 5 year old Operating System

Not only is it a 5 year old but also not longer supported by Apple for updates and probably any security fixes.

Generally, upGrading to a newer version of macOS is painless

As long as the user has not installed Third Party Security Software, Commercial VPNs, Third Party Disk Cleaners and Third Party Two Way Firewalls

Have a 3-2-1 Rescue Plan

If you have used or are using Time Machine Backup Utility  you maybe able to Restore the Deleted Files 

For future purposes

To truly protect your non replaceable Data

Have a 3-2-1 Rescue Plan in place and always current

3 Backups using 2 methods and 1 off site incase of natural disaster or un-natural disaster.

Each of the above should be done to a Dedicated Single Purposed External Drive 

Below link is intended to augment what TM Backup does 

https://bombich.com