But visibly that was the only way to (try to) convince Cersei to do a truce, if not a temporary alliance, given that nobody south of Winterfell believes that the White Walkers are real. The idea is sound, and was indeed more of a "surgical strike" to capture 1 walker, instead of sending a large troop. As for the question of why going himself, he believes that his knowledge of WW combat makes him valuable (he is correct in that) and that there are other people who could take over if he fell (also correct).
Now, visibly, the Night King has informations that Jon doesn't and was expecting them, and expecting a Dragon rescue too.
My theory is that the NK knows of a detailed prophecy regarding the fall of the Wall, that allows him to have knowledge of things to come, that's the only explanation.
Which in thereof doesn't mean that Jon is an idiot btw.