General information on the mission
The main objective of the eleventh mission in Metal Gear Solid V is to defeat Quiet, i.e. a member of the Cyber organization, who is armed with a sniper rifle. There are two different ways to succeed - by depriving Quiet of her entire health bar in read, or by draining her endurance bar in blue. After you succeed, the game will present you with an important choice to make - whether you want to kill Quiet or to arrest her.
PRIMARY (1/2): Eliminated Quiet - You can defeat Quiet by depriving her of either the entire health bar or the endurance bar. In the case of the former, you do not need to worry that you prematurely kill her, because both variants end in the same cutscene.
PRIMARY (2/2): Determined what to do with Quiet - Regardless of the way in which you defeated Quiet, towards the end you need to decide whether you want to kill her or to take her hostage. The better variant is, of course, the latter one, because you gain a new Buddy this way.
SECONDARY (1/2): Neutralized Quiet without hitting her with lethal weapons - You do not need to resign of using weapons completely, but only the non-lethal ones are permissible, thanks to which you will be able to restrain the target )e.g. sleeping ammo pistol). Also, you need to make sure that you do not use grenades, or other explosives.
SECONDARY (2/2): Neutralized Quiet with non-firearm attacks - The easiest way to do this is to set a supply drop onto Quiet. Alternatively, you can also surprise her in melee.
You do not need to decide for the night, in the case of this mission, because even then Quiet will have no problems spotting you. Apart from that, it will be easier for you to find her during the day, after she moves to a different spot.
If possible, take the D-Dog along, who will help you locate the sniper. As for your equipment, if you are going to deprive Quiet of her health bar, get the best sniper rifle available. If you want to deprive her of her endurance bar, your choice of weaponry plays no major role.