How do I change the stock firmware in Samsung Galaxy S3 Neo to higher than 4.4 KitKat without rooting? My Samsung Galaxy S3 Neo has Android Kitkat. Is there a way to upgrade (without rooting the phone) to a newer version?
Quite simply, no. Samsung support ends at KitKat due to its age.

There does seem to be official CyanogenMod 12.1 (Android 5.1.1) and even unofficial ResurrectionRemix (Android 6.0.1) available, but you will need root and/or recovery to flash them, so I'll not go into details.