Q: how do I upgrade safari when it is not on the available update list

If you are using OS X Mountain Lion 10.8.5 as indicated in your signature, the latest version of Safari that you could install is Safari 6.2.8 and it is no longer supported.

Your options are to upgrade to a later OS X or use a browser that is still supported. The current version of Firefox is still supported but that is also coming to an end: https://blog.mozilla.org/futurereleases/2016/04/29/update-on-firefox-support-for -os-x/

You can run the latest Safari version 9.1.2 if your Mac can upgrade to El Capitan.

Check the requirements to install El Capitan here > Upgrade to OS X El Capitan - Apple Support