In an honest hearing, I think Cruz would lose but the SCOTUS is good at going into weasel mode and avoiding their responsibilities. They would simply avoid making a decision until it was irrelevant.
They'll evade the issue, most likely. Might be tough, because I foresee some liberal judge seeing an opportunity to be a hero for the liberal cause, ordering a Secretary of State to remove Cruz's name from a ballot. It only takes one, and there will be literally hundreds of cases before this plays out.
I think it is best to have Cruz in the race, and for him to strongly resist attempts to disqualify him, even by procedural arguments. Awakening the public is hard, maybe impossible, but for it to have a chance, the debate has to become hot in the minds of a sufficient fraction of the public.