Before going in
Be prepared: Have your standard Priest raiding loadout ready. In addition to your flasks, food, and pots consider packing along some Light Feather in the event you get to Archimonde. It's use will be explained later.
Be prepared - mentally: I said it earlier but I'm going to say it again. There's a reason why this raid instance is the most punishing. You will face waves and waves of Undead. If your raid wipes at any point, you start back at the first wave. Believe me when I say it can be mentally taxing.
Shadow resist: You might find having a few pieces of Shadow Resist helpful. I went in with a Medallion of Karabor and Night's End. Combined with the Shadow Protection buff from Priests, it added up to 150 Shadow Resist. With luck, you've been able to do the quest required to get the neck.
PvP Trinket: Get your Medallion of the Alliance or Medallion of the Horde to help break the Icebolt spell from Rage Winterchill and to break Fear from Archimonde.
Shackle Undead: There are a few waves where you need to use your shackles. Your raid leader should be the one determining what gets shackled, but shackle duty should go to Shadow Priests since they're likely to have some hit. Prioritize Banshees over Abominations for the waves where both show up. If not, shackle the Abominations if the raid needs it done.
PvP Trinket: If you get hit with an Icebolt, this is the time to use it. This frees you from the trap.
Death and Decay: Circle of Healing Priests, get ready to spam that spell on melee DPS who get caught inside.
Positioning: Try to space yourself and the rest of your healers around Anetheron in equal intervals around the boss.
Shadow Resist: Equipping the cloak and the neck allows a chance to mitigate some of the damage done to you.
Mana management: Don't go overboard with heals right away. Ensure you have your mana regeneration buffs and consumables ready. Try to stay above 3000 mana as much as you can. If you end up below 3000 mana, turn around and run away from your raid before you explode.
Shadow Resist: Equip your Shadow Resist gear. It's got a chance of resisting the silence effect which I've found to be quite handy.
Trinket: If you have a Vial of the Sunwell on you, it's another excellent trinket option for those desperate situations when your tank is low and you're unable to get a heal off.
Don't forget your tears!: Talk to Tyrande and get the Tears of the Goddess. However, you can be like me and cheat and carry a stack of 40 Light Feathers. This allows you to Levitate down in case you use your tears too early.
Fear Ward: Work it out with the other Priests in the raid to figure out a Fear Ward and trinket order. You want to ensure there is a healer available to help support your main tank on Archimonde until the fear breaks. If you want to be real fancy, try to time a Mass Dispel on your group to coincide with the fear.
And that's all there is to it! Hopefully these tips will serve you well when you work your way through this instance. With luck, you won't have to go through the walk of shame too much (it's what I refer to the run back from the graveyard). May your loots be phat!
Want to find more great tips for carrying out your Priestly duties? Spiritual Guidance has you covered -- before the long hiatus, we looked out an introduction to playing a Priest and different types of healers. And don't forget to check the WoW Insider Directory for more priestly info -- there's lots of guides, writeups and discussions over there as well.