Winning the duel with A- rank
If you just want to win the duel and you do not care if you obtain the highest S-rank, you can utilize the below tactic. It will allows you to win within 1-2 minutes, without the risk of engaging Quiet in a direct way.
After the duel starts, do not lean out from behind the wall where you should be hiding by default (remember that only one hit may end in your death!). Instead, equip binoculars and locate Quiet. Mark her position.
This tactic includes using sleeping gas against Quiet. For that, access iDroid and open the Missions tab. Select Fire Support and then, Sleeping Gas. As the spot to raid, select Quiet's current position (the above screenshot).
Note - A single raid is 30000 GMP. Still before you start the mission, make sure that you have enough cash and can afford, at least, two raids.
Do not move, to prevent Quiet from changing positions, and wait for the raid to start. Her blue bar, as a result of successive raids, will start going down. Unfortunately, probably you will not defeat her with one raid only, because sooner or later, Quiet escapes into another spot of the ruins.
Now, it's the most difficult part. You need to move into a different spot and locate Quiet again. Sprint from one cover to another and make short stops behind each one. You cannot let Quiet take a shot towards you, because it is highly possible that you will get shot and die.
After you determine her approximate location, use binoculars to locate and mark her again (here, you can be helped by the D-Dog). Just like before, call for an air raid with sleeping bombs and remain behind cover for the whole time. This time, the raid should deprive Quiet of the entire bar and the battle will end (towards the end, you can decide again whether you want to spare or kill Quiet - pick the former).
How to win the battle with S rank
Ranking S requires you to fight with Quiet an entire battle. You should try for the rating only after you have progressed very far into the game and you obtain the best gear available.
Before you start the battle with Quiet, it would be a good idea to get prepared properly/ Take along the best sniper rifle you have. It would be best to take a model with conventional ammo, thanks to which you can deplete Quiet's health. Apart from that, you can make preparations that will ensure you with surviving Quiet's attacks. From the briefing menu, select the Characters tab and find someone with the Tough Guy ability (the above screenshot). This type of characters have increased resistance to damage. This is not all, because such beefcake needs be geared with the best combat armor (Battle Suit) and a headgear. If you ensure all of these, this should guarantee that the character will survive after a shot and only takes severe damage.
Right after the battle starts, you need to learn dealing damage to Quiet, over again. Always start by leaning out from behind the cover, for a fraction of second, which should provoke Quiet to taking a shot. At the moment, at which she is busy reloading her rifle, lean out and take your shot. Of course, you cannot lose time on accurate aiming, therefore always face the right direction and take a shot always when you see any of her body parts in your sights (not just the head).
Note - In extreme situations, Quiet cannot may not know your location. Then, you can, of course, take your time and take a better aim.
Quiet, each time she takes damage, moves to a different spot. Moving around the map is yet another factor that you need to master. Dash over short distances and always choose your next cover beforehand. Additionally, you can end each dash with a lounge to the ground, which minimizes the risk of Quiet's attack. To locate Quiet again, use binoculars and D-Dog (unfortunately, the dog can locate her at a distance no longer than several dozens of meters). After you find Quiet, find a nice little spot to take another shot. Repeat this pattern until you deprive Quiet of her entire health and do not take too much risk, especially if you are playing as a character that can die after a single shot.
Note - In the Extreme battle with Quiet, the option to drop supply drops onto her does not work. Therefore, do not waste your cash, because Quiet manages to escape before the crates drop onto her.