A 3DS Update is not the same as a Game Update
By updating your 3DS system, you get bug fixes, eShop content, etc. By updating your ORAS game (or any game), you get all of the patches and bug fixes for that game. This is how you will be able to access the PSS. To update your game:
1- Turn on the 3DS system (duh)
2- Open the eShop
3- Type your game into the search bar
4- The first option that comes up should be something like "Alpha Sapphire Version 'X'" (let X represent the update number)
5- Select the Update option and proceed to update
6- Wait a few minutes for the update to finish
7- Open your ORAS game and use the PSS
Enjoy your PSS access, compliments of Nintendo! (And moi.)