Upgrading Sierra

What model year is your 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

Yes, you would need to upgrade to get a later version of Safari.

Be aware that if you upgrade to Catalina or later any non-Apple 32-bit apps that you have will also need to be updated. See: https://www.macworld.com/article/232735/how-to-check-if-mac-software-is-32-or-64-bit.html

There are some recommendations not to upgrade directly but Apple upgrade instructions don't include that.

See the following from How to get old versions of macOS - Apple Support

Safari uses these links to find the old installers in the App Store. After you download the installer from the App Store, the installer opens automatically.

• macOS Big Sur can upgrade Catalina, Mojave, High Sierra, Sierra, El Capitan, Yosemite, Mavericks

If you want to upgrade to High Sierra first then you may want to contact Apple Support.