You need to have sea access (for Limbus), then talk to Sagheera in Port Jeuno (near the Windy airship docks entrance). She'll tell you about upgrading your AF, as well as other things you can spend your ancient beastcoins on. For this particular AF+1 piece, you trade her Diabolic Silk (rare/ex drop in Limbus), Diabolic Yarn (rare/ex drop in Limbus), Ether Leather (synthed with 91 alchemy; Dried Marjoramx3, Ram Leather, Ahriman Wing, Treant Bulb, Distilled Water w/Water Crystal; can be bought on AH), 25 Ancient Beastcoins (common drop in Limbus, buyable from bazaars), and your original NQ Wizard's Petasos. It takes some time for her to make it, but I'm not sure how long exactly.
Good info Kyrial, thanks ^^ Now I just gotta figure out what AF2 I want, what AF1+1 I want, and what other armor pieces I want so I know what to spend Ancient beastcoins on.