All classes have the ability to solo.
The priest, is one of the more challenging classes to level.
Priest's largest weakness in the unfortunate defense of cloth armor. Although this armor can be supplemented by using light/medium/heavy leather armor kits. This will still not provide them with the defense needed to handle direct melee combat without a good amount of spirit (Magic points, mana etc..)
In melee combat the priest is weakened again by the use of only low damage melee weapons, such as daggers, maces, staves & wands. This is most devastating if your spells are being resisted by the mob you have decided to test your skills against.
Arguably the most useful spell caster in the game, a priests spells made it necessary for Blizzard to give priests poor quality weapons and armor. Priest spells range from NPC control (PC control in PvP i think?), protection spells which all the priest to be kept clear from all harm for "X" amount of time, to high DPS spells that devastate the most hearty adversary. Should any of these spells be resisted though, the priest’s, challenge is to stay alive while its armor and weapons (and low hit points) do the work.
My over all assessment is that even though leveling is far easier with a hunter/warrior/rogue. Your time as a priest is more interactive and provoking then what could be a simple click to attack class.
(Hunter is my favorite class right now, but priests are a lot of fun)
Thanks for your time.