I don't think we know that yet. From what it sounds like, yeah, I doubt it'll be any sort of traditional election, but it has been indicated that he will have the responsibility thrust upon him even when he doesn't want it. Sounds like he'll be appointed. And based off the fact that Supergirl will be president years down the line, I imagine it's a position that the world will recognize.
Based on what we know so far from the run, I'd say it's likely gonna be a little bit of both. Jon's character arc has been about the idea of him knowing that one day people are going to expect him to be That Guy because of who his father is. The responsibility and expectations that come with such a highly regarded legacy progenerated by The World's Best Dude. So, in that sense I'm pretty sure the passing of the title will be something akin to the passing of a crown.I hope that bendis shows jon to earn the title from the people like his father.
But you can be sure Bendis will play up how some people aren't down with it (be it played for jest or seriously). He's done the same with Miles and Riri. But unlike Miles and Riri, Jon is coming into being Superman with a pretty stacked resume as Superboy on Earth, then in space, helping people on Earth-3, surviving Earth-3, coming up with the United Planets (I imagine Clark will mention that at least.), and serving as part of the Legion. How much of that resume along with his deeds to come will win people over remains to be seen.