Priest has always been mid-tier in Hearthstone Arena. The class was mostly dominated by other classes which have always outshined it due to the cards they received. Priest has had better luck in Constructed. Priest has always had high power levels with its Basic and Classic cards such as Mind Control, Northshire Cleric, Shadow Word: Pain and Death and many others.
The expansions Journey to Un'Goro, Knights of the Frozen Throne and Kobolds & Catacombs will rotate out from Arena in 2019. Classes such as Druid, Hunter, Mage and Paladin are losing some great Arena Cards from their pool too. Here are some notable ones from Priest.
#5. Shadow Ascendant

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Expansion: Knights of the Frozen Throne
Shadow Ascendant is a vulnerable 2-drop but is a great buff to minions in your board. The +1/+1 stat to a minion can make a real difference in Arena and high health, high attack minions have a lot of value.
Shadow Ascendant doesn't have to be played on turn 2 alone, it can be played on later turns when you have a stronger board. When you hide this behind a Taunt it starts becoming a threat, especially if it buffs your Taunt.
#4. Obsidian Statue

Expansion: Knights of the Frozen Throne
Obsidian Statue is arguably much better in Arena than it is in Constructed. If you play this on turn 9 in Constructed it can be really slow. But a turn 9 Obsidian Statue play in Arena can be life-saving. Obsidian Statue and the Warlock card Voidlord are the best taunts in the 9 mana card pool.
The Deathrattle of Obsidian Statue can be disruptive to your opponent and make them lose a lead in the board. If they hit the minion with their minions, you can Lifesteal a lot of life back. It is a solid taunt that will be a big loss for Priest.
#3. Crystalline Oracle

Expansion: Journey to Un'Goro
This little guy has the same tier score in Heartharena as Obsidian Statue! Crystalline Oracle has the same effect as a Thoughtsteal but the 1/1 body is useful to play on turn 1. Even though it is very RNG based, the value of a card from your opponent's deck has more value in Arena. You could get something useless like a Deadly Poison sometimes too, but even copying a Neutral minion is useful.
#2. Free from Amber

Expansion: Journey to Un'Goro
Free from Amber is a spell that has so much value in Arena. You can discover an 8 cost to 10 cost minion for a potential discount. You can summon a Taunt to keep you safe, a giant dragon such as Ysera or anything else to help your situation and board.
Most minions that cost 8 and above, including the giants, are pretty well statted and there are very few bad ones. You might sometimes get a Togwaggle but the Discover mechanic gives you better odds with the other two.
#1. Psychic Scream

Expansion: Kobolds & Catacombs
Psychic Scream is probably the best board clear Priest has received in Arena. It is one of the best cards in Arena. Cards like Violet Wurm are very common in Arena and if you psychic scream those 1/1 guys, you will restrict your opponents' draw by a lot.
Draw mechanics and deck thinning are barely present in Arena, so if you shuffle a lot of minions into your opponent's deck, especially weak token ones, your opponent has a hard time drawing the right cards.
Honourable Mentions
Curious Glimmerroot

Expansion: Journey to Un'Goro
Curious Glimmerroot has an interesting "guess" mechanic and its power level is similar to Crystalline Oracles. The card choices are always Class Cards. You might be able to guess it if you saw your opponent play that card or if you played enough to know what's common. It has a fairly statted body to go with it too.
Shadow Visions

Expansion: Journey to Un'Goro
Shadow Visions is just as good in Arena as it is in Constructed. You can discover copies of Mind Control, Holy Fire, Free From Amber and other extremely high power level spells if you drafted them. Priest has a lot of useful removal mechanics and it is highly likely to draft a good spell. Shadow Visions will be a hard-hitting loss for Priest.
Devour Mind

Expansion: Knights of the Frozen Throne
Devour Mind adds 3 different cards for the cost of 5 mana. In Arena, discovering copies of synergizing class cards or even neutral minions is very helpful. If you run out of cards in your hand, you can recover 3 more cards with this spell. It has the element of RNG to it, but it is a pretty good draft.
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