My Name is Zushio(1999)

Yo no Na wa Zushio

From Anime New Network:
Zushio was to be king and promised his sister that he would become king (on penalty of death). Unfortunately for him, his empire was overthrown and now he's a wanted man. A girl has the misfortune of running into Prince Zushio and becomes tangled in his plight. Adding to her misfortune, Zushio is not quite right in the head, but hey, at least he seems to be immortal.