Is there really a story in the Pali Canon about a layman running away from home AFTER having attained arahantship?
As you found out on Reddit, the story sounds a lot like that of Yasa, who didn't of course become an Arahant before meeting the Buddha, and certainly not in his sleep! But to your general question, stories of lay people who have become arahants and then left home do exist in the texts. A notable example that comes to mind is Khema Theri (see palikanon.com/english/pali_names/ku/khemaa.htm).

Worth keeping in mind is that an arahant needs to learn from a Buddha to become enlightened, so if the story in question means the prince became an arahant before meeting the Buddha, that is impossible.