Software Update

This is not an ATT phone, has nothing to do with ATT.  Whether or not you can download apps again is entirely between you and the google App Store and your purchase history.   

It's definitely an AT&T phone. I bought it through AT&T, my provider is AT&T, and the update is an "AT&T Software Update." It's telling me it needs storage and to delete stuff, and what I'm asking is if I'll have room after the update for everything I deleted. 

AT&T never branded any Duos version of any Samsung phone.

The model number for an AT&T J3 will be the J320A for the 2016 version of J327A for the 2017 version. When you go to Settings and About Device/Phone what model number does it show? Because if there is no A designation after the model number, it's not an AT&T version.

It is the J320A, but will I be able to put everything back onto my phone afterwards? 

Duos usually indicates a 2 sim phone.  ATT has never sold a phone with 2 SIM card slots.  

Maybe ATT Mexico did.  

Doesnt matter.  No matter where the phone came from.  Your and data restoral are nothing to do with the carrier.  Typically music and photos are what needs to go.