Magician Faust Sidequest
This quest is tied in with the Stardust Sidequest in the game, in order for you to challenge Magician Faust to a fight, you'll first need the complete the Stardust Sidequest which will reward you with The Vanquishing Stone, the item required to challenge Magician Faust to a fight.
Once you've obtained the Vanquishing Stone you'll want to travel to the Tower of Flanvel which is where you faught Lloyd for the second time earlier in the game. In the screen shot above you'll see exactly where you should go in order to reach Magician Faust. The first time through this area you had to run north past the save point but this time you take the teleporter on the right.
Before you challenge Faust to a battle it's strongly recommended that you purchase Legend Casques from Lohan since they'll decrease the amount of spell damage that you take from each of his attacks. This is almost a requirement if you want to defeat Magician Faust without spending hours upon hours grinding.
Magician Faust is an extremely tough battle, maybe even harder than the final boss himself. Each time he attacks, he typically gets four or five combat rounds, wailing away at your party with an assortment of different spells which will deal a buttload of damage if you don't take the proper precautions.
If you don't have a Legend Casque on each character (or at least one or two characters) it's strongly recommended you bring many Healing Potions as well as Meru or Miranda, someone who is capable of healing your party with Dragoon powers. You'll be taking a LOT of damage during this fight and you'll probably have someone in your team healing each round.
Also, it's recommended that even without the Legend Casque armor you don't enter battle and use your Special on everyone, transforming into Dragoons. If you do, you'll be unable to use items which greatly decreases your healing effectiveness.
Reward: Phantom Shield (Decreases damage from all attacks by half)
In addition to the Phantom Shield that's rewarded to you after beating Faust, you'll be able to pick up three chests from the room too which contain the following items...
Holy Ankh (Revives from death with a given probability)
Magical Hat (Raises Maximum MP by 50%)