Q: Can I upgrade my outdated Safari browser without having to move to El Capitan?

No. You would have to upgrade OS X. You might be able to get away with Chrome if Google still supports 10.8.5.

Unfortunately not, switch to Firefox if you don't want to upgrade OS X

If you can upgrade to Mavericks, which would be possible if you have ever downloaded it, you could use Firefox with at least some confidence that you will be able to keep it updated. I believe that OS 10.9 Mavs is going to be the oldest supported OS once support is dropped for earlier OS versions within the next year.

Why don't you want to upgrade to El Capitan, it's better than Mountain Lion and more secure?