As far as your S3, system updates and patches would get applied when the carrier released them. That update menu in your Settings would rarely do anything, especially for Samsung branded phones. It's an item that probably wasn't even necessary to add into the menu options, way too many people have been deceived into thinking it would actually do something. Also, the S3 has not been supported for years now (it's coming up on five years old, quite dated as smartphones go) so system upgrades won't be very current in any case.
If you really, really want to get a newer version of Android on your S3 you'll have to root it and then install a third-party ROM. But if you don't want to go through the process of rooting, you might not be able to get it running anything newer than 4.1.2, some carriers supported S3's up to KitKat 4.4.x, some stopped at Jelly Bean 4.1.x. There is a small chance you can manually upgrade your S3 using an OEM ROM but it's very dependent on the exact model I.D. of your phone.(Samsung ROMs can be applied without rooting, it's the third-party ROMs that require root). What is the exact model name of your S3?