Google Play app doesn't auto-update Ever since the new update of Google Play app, I couldn't get the apps to automatically (in background) update.
But instead, they only update, when I open the Google Play app a few times

I have auto-update and notificatoins checked. I have auto-update apps ofer WiFi only set.

I have JB 4.1.2 and Galaxy S3 GT-I9300.
I think any auto-update/notification settings relate to apps installed via the Play Store, not the Play Store itself.  That's updated occasionally by Google.  You could try and force an update, if you're convinced one is available and that you're missing it, by going into the Application Manager and uninstalling updates for Play Store, then clear data, and clear cache, then running the app again and seeing if it forces the download of a newer version.

(Incidentally, I have the S3 too, and am currently on v4.1.10 of the Play store. Which version are you on?)