The priest is not vulnerable until his name appears above him. This is generally just after he touches down on the ground. The fireteam should spread out and go for cover. This will keep him from just blasting the team away.
Like any of the Archons, this one comes in with a Sharpnel Launcher. He will make good use of it. Be sure to have a good spread of the team and someone ready to revive whoever drops.
After you get a portion of his health down, he will call in help. This brings in a number of Fallen. You will encounter a number of Vandals and the odd Captains. The Fireteam should work to have at least 2 people on the boss while the final person works to thin out the Fallen and help any of the Guardians that get taken down.
Toward the end of the battle there will be a High Servitor that comes in. This is a Major variant of the Servitor. This orb boosts the defenses of anything nearby it with a purple aura. Taking it out quickly will work in your favor. Remember to aim for the eye in the front.
Defeat him and you will be done with the strike!