I don't think the stock OEM apps auto update. If you want to check for mass updates in the Galaxy Store, you have to have a Samsung account. If you don't have a Samsung account (or you don't want to create one), you have to go to each app, go into their settings/about, and then it will check if there's an update or not. Eg. Phone app, Settings, About Phone.
Also, make sure you update the phone to the latest available firmware. The carrier can be an issue as well. Try different carriers, or switch to 3G. There are too many variables. Eg. kind of apps you have installed, whether the phone is rooted or not, etc etc. Worst case is to reset the phone to factory and see if the lag is still there.
However, in the end, it's an old phone. Any smartphone using eMMC will lag as eMMC cannot do read and write to the storage at the same time. It can only do one or the other, meaning simple things in a multi-tasking OS like Android can make it crawl really quickly. This is why I would avoid any phone that is still using eMMC. I feel it's time for you to upgrade. I'd suggest the Galaxy A52.
