I think this is the version of Arthas you control in the quest The Hunter and the Prince, because you get to control "Prince Arthas Menethil", and this is the only NPC on WoWHead with this name. Also, he has the same model as The Prodigal Leader, which is basically the Lich King without the helmet, and he looks just like that in the quest.
Elite
Prince Arthas Menethil
Prince Arthas Menethil is an Elite Undead, level 25 to 30, friendly to players.
Details
Prince Arthas Menethil at a Glance
Level and Map Tracking
The game data tunes Prince Arthas Menethil for levels 25 to 30, Wrath of the Lich King content.
In the Client Files
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Arthas killed his father and became king of Lordaeron. If you kill the Lich King means you are the new king of Northrend. (And maybe new owner of Lordaeron)You can even become the king of northrend but in no way that means "owner of Lordaeron". Arthas didnt became king of lordaeron by "killing the actual king and reclaiming that position". He bacame king since he was the prince, and the actual king was mudered (by arthas). If you kill Arthas you would just be a killer. That wont give you the right of the throne of lordaeron.
Ner'zhul was defnitely a shaman.
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