Jadis tries to manipulate him into revealing the location of any other humans. Then she tries to torture him into revealing the location of any other humans. Once both tactics reveal that Joffrey has no idea how he got here or how to reach other humans, she kills him.
He might get the Cousin Eustace treatment and leave a better person. Or get stomped to death by tiny goat legs ten minutes in.
Couldn't the prophecy work to Joffrey's advantage, where the creatures of Narnia have been waiting for a "Son of Adam" to liberate them from the White Witch for decades? How long before the creatures of Narnia get tired of his attitude? Though if Joffrey meets either the White Witch or Mr. Tumnus right away, he would likely be screwed. Mr. Tumnus spared Lucy due to kindness and innocence and would have no reason to do the same to Joffrey, while the White Witch would easily kill or manipulate him.