I guess he would become captain of the guard if he (or others think he) deserves it.
He could be appointed King by the Council of Elders if he is heroic like his father was.
But Adam (not He-man) doesn't have a good track record for being a hero...
EDIT: Maybe MOTUC Adam isn't acting as a carefree prince. The bio may shed a light on this... IF Adam gets a bio now that he is a Orko pack-in
EDIT2: Miro has gone missing according to Randors bio, he never was pronounced dead. That didn't stop the Elders appointing Randor king.