Ironically (to my last comment), I just faced 4 Priests in a row, all Deathrattles.
I assume that these were N'zoth priests, but again, they never managed to actually get N'Zoth online.
Quest Shaman pretty much ran them into the ground before they could get going.
The key is to basically hold off attacking in the early game, so the priest doesn't get cheap healing to advance the quest.
I feel like quest may not be the way forward for Priest. It can be a struggle to make quest progress against anything that doesn't rely on dealing incremental chip damage in the early game, and even though there are a couple of little tricks you can use to advance the quest in a slow game, ultimately you are at the mercy of the opponent's decision to interact with you if you want to complete the quest in good time. I could be wrong however - completing the quest quickly in a slow matchup is likely of very minimal importance, as you're relying more on setting up a good N'Zoth pool and winning in the late game resource battle than giving some taunt minions a small stat boost and some extra stickiness.
Oh I mean as quest priest I ussualy complete my quest around turn 8. After you can start dropping the big boys enemy must start trading. Imagine dropping a 2/6. Enemy must contest it, then you ressurect it, and it ussualy is not possible not to leave any damage undone.
In control matchups it's even harder to complete quest. Quest priest Vs quest warrior though key is your hero power. If you get quest complete, warrior ussualy runs 4 so how target removals. You accept that, after seeing all 4 of them, if any minnion sticks you boost it up to levels where warrior must brawl to remove it. Eventually you win. If he brawls too early you mass ressurect or nzoth. When it goes to fatigue you seanse his elyssiana and win fatigue too. Priest wins Vs warrior perfectly. Unless off he gets Dr boom very very early while you don't develop your hero power.