You choose the target of the first ability before paying any of that ability's costs. In other words, you can't choose Priest of Fell Rites itself as the target to bring it back from the graveyard.

Activating a creature card's unearth ability isn't the same as casting the creature card. The unearth ability is put on the stack, but the creature card is not. Spells and abilities that interact with activated abilities (such as Stifle) will interact with unearth, but spells and abilities that interact with spells (such as Cancel) will not.

If a creature returned to the battlefield by the unearth ability would leave it for any reason, it's exiled instead—unless the spell or ability that's causing the creature to leave the battlefield is actually trying to exile it. In that case, the spell or ability succeeds at exiling the creature. If the spell or ability later returns the creature card to the battlefield (as Ephemerate might, for example), the creature card will return as a new object with no relation to its previous existence. The unearth effect will no longer apply to it.