The Basics:
It is important to understand that a hunter/mage/priest team requires synergy, and a lot of it. The strats that you need to execute in order to win require well played timing and teamwork. Communication, especially between the hunter and mage is key. There should never be a moment when your vent channel is quiet upon joining a game. I encourage you and your partners to never shut up! You'll understand better once I go more in depth about the strats. My partners are very good friends of mine, we understand each other's play styles and know how to support one another. Think of your teammates as extensions of yourself, that you're just a part of one big well oiled machine because things will have to go perfect. One misplaced scatter shot or frost nova can and will be the difference between a win and a loss.

The beauty of this team makeup is that you are very flexible. You are first and foremost an outlasting team, but you have amazing burst when you need it. The majority of your time will be spent kiting around pillars, and utilizing all your hunter's abilities in conjunction with your mage to protect you and your teammates. Patience is important, if you lack patience and just want to blow stuff up then this combination is not for you.

You outlast by keeping your opponents damage output to a minimum with sheep + frost nova + traps + kiting + scatter and mana draining via viper sting and scorpid pet. You and your mage have to protect your priest at all costs. You need to make sure melee does not have a chance to touch him. Casters should not be able to get line of sight on him to cast. If the other team focuses your hunter or your mage instead, same thing goes.

The Beginning:
Among your priest buffing and mage summoning water, you should be getting raid targets set up (Circle, thong/upside-down nacho, moon, etc). You'll need to know where your partners are at all times. My Priest usually will put Fear Ward on me off the start, and then on himself for the rest of the game once the CD is back up. This ensures me being able to start my draining immediately, regardless of a fear bomb, but then again, he's undead and has WotF, so perhaps this wouldn't be ideal for a non-UD priest. Use Aspect of the Viper.

The arena starts, the gate opens, immediately the hunter should be checking track (A tracking mod such as ICU and an arena mod like Proximo or Arena Unit Frames will make the job a lot easier) to see what you'll be facing. Then label your enemies: There is the Drain Target and the CC'ed Target. They obviously cannot be one in the same because viper sting breaks sheep. There is a third target, which is sometimes the drained target as well. This is the enemy we're constantly applying pressure to. We do so via instant attacks while kiting.

Depending on the arena, you'll move to your kiting spot. On Nagrand, we usually use the pillar to the left upon entering. On Lorderon, its the side of the coffin closest to the green puddle of sludge-poop stuff. On Blade's Edge its on the ground level at the closest pillar to our starting room. Be patient and stick to your spot, let them come to you.

On the Art of Draining:
The first thing you do upon seeing what your opponents are is determine who you'll be mana draining (The drain target). It's not always the healer as one might think. We put priority on shadow priests/mages/elemental shaman, etc before we ever drain their healers. Their DPS is the biggest threat, and mana-based DPS classes are worthless when they're oom. Improved Stings in the Marks tree and a Scorpid pet will make your job easier.

Set your drain target as your /focus, this makes it so you don't have to break from your current target when using the following macros:

/cast [target=focus] Viper Sting
/cast [target=focus] Silencing Shot

Every 15 seconds you need to be reapplying viper sting, until they are oom. Be sure your scorp is always on them and watch his health. Don't let the pet die or the game gets drastically more difficult as your drain target will have a lot more opportunities to drink. Also, if your priest is confident that he'd be safe in doing so, he may help you drain by throwing a mana burn on the drain target, but he should never do so if he or his partners are in danger. Healing > Mana Burning. You'd be surprised at how many priests don't understand that concept.

Tip: If the other team is trying to kill your pet, instead of wasting your priest's mana trying to keep it up, dismiss it. Bring it out again later.
Tip: When draining pallys follow up your sting with a silencing shot to buy another tick or two of the drain.

The Second Step, and a note on Rogues:

When the game starts, lay a frost (slowing) trap down at your kiting spot. There should never be a time when that trap isn't down. Have your partners be aware of where its is at all times, this is extremely important. Drop a flare if there are stealthers; make sure you and your partners camp that flare. Determine if the stealther is a druid (Check if the visible enemies have MotW), if not then attempt to hunters mark a non-stealther (to help watch their location) and turn on Track Hidden. It is extremely important that you do not let that rogue get a cheap shot off on you or your partners. Keep flare and frost trap down, have your mage rank 1 arcane explode between flares. Spam tab and scatter shot to flush it out before given a chance to get the opener on you.

Once the rogue is exposed and the battle engages, make sure you keep a flare down or that you're fast on casting another one once that rogue vanishes. Tell your partners to call out in vent the second the rogue vanishes because if left alone, rogues will shut down your entire operation single-handedly. Rogues lock down their target making it impossible for you or your partners to kite out of line of sight. They expose you to being burnt down and that is why we don't let them have an inch of freedom. When there is a rogue, they are always the CC'ed target.

The Chase:
As stated before, it is best to be patient and let them come to you. After you set your focus to your drain target, switch targets to the enemy that you and your mage are going to be keeping locked down (Your CC'ed target). For example's sake, let's say a warrior. Wait for him to charge in and immediately get a scatter off on him. This is when my mage sheeps him and we begin the kiting. Again, keep that frost trap down on the pillar you're kiting around. We kite to break line of sight and to protect ourselves. It keeps us in control of the situation.

My mage will continue to sheep him until he gets diminishing returns. The second he is not able to sheep or frost nova that target, it is the job of the hunter to wing clip and scatter him, buying time for my mage to be able to sheep once more. You can not let the CC target have any freedom so save your scatter shot for that target and that target alone.

Between scattering the CC'ed target when needed, laying frost traps, keeping flare down (when needed), and draining your /focus target, you and your mage will be damaging one of the two non-CC'ed targets with instant attacks to wear them down while kiting. If there's a lock on the team, we usually DPS him so we can keep him from tormenting our priest.

I'm going to put a bit of emphasis on the point above, as it didn't seem to be very prominent when it should be. You must do medium DPS to whichever target you make your DPS target, which will in most cases be your drain target. Stay safe, and don't open yourself up to being blown up or shut down, but you need to be applying pressure on your DPS target. This doesn't mean go crazy on the guy because most likely if you do, you'll run yourself and your partners OOM too early, or you'll forget your other duties (traps, viper, keeping your teammates safe). Use your better judgement.

Tip: Make sure not to break sheeps with Multi-Shot.

Victory:
Keep it up until your drain target is oom. Start to drain whatever else uses mana, keeping your pet on the first target. It is now safe to get out in the open and start to burn the target of your choice, which is usually the poor sap who's oom. Rapid Fire an Aimed Shot on (for the Mortal Strike effect) and you and your mage go in for the kill.

Don't get so caught up in the bloodletting that you forget to keep your CC target CC'ed and your traps down as your priest will most likely still be LoS'ing behind the pillar. You must keep the sheep controlled until the very end. Finish them off one by one, but don't get too cocky, a lot can still go wrong. You're only safe out in the open when their DPS'ers are neutralized via mana drain and sheep, so keep it up.

Final Notes:
What I've given you here is only the basic skeleton of a strategy. Obviously there are going to be instances that you won't be able to execute the above plan of action effectively, but that's the beauty of this team. We can adapt at the drop of a hat. Over time, you'll learn when to follow it to the letter, and when to improvise. Remember, you're nothing without your partners. If one of you messes up once, it's usually over. Get to know them, their play style, their strengths and their weaknesses, and support them. Trust me, it's frustrating at first, but just be patient. It's not an easy combo to pull off, but once you do, there's very very little that can stop you.