What First Amendment challenge? Since when does the First Amendment oblige the school to give the reporter credentials? Methinks you have no clue what the fuck you're talking about. You do realize that getting credentials is a privilege and not a right, correct?
If that "privilege" is conditioned (by a state institution like a public university) on the content of the coverage, it would absolutely be a First Amendment violation. if the reporter could prove that the reason for denial of press credentials is the content of his reporting, he would win if he brought suit. It's not even a close question on the law; it's proving the necessary facts that might cause problems for such a plaintiff.
In this case, it's a closer question of law. On its face, the restriction is content neutral, and would probably be judged as a time-place-manner speech restriction. Those can be upheld, but they're typically upheld when the speech itself impinges on an important state interest. For example, amplified sound in the park interferes with the state's purpose of allowing citizens quiet enjoyment of the park. As long as the ban on amplified sound is content neutral, both on its face and in the manner in which it is enforced, such a restriction would probably be upheld. In this case, I think it would come down to whether the court judges the revenue from contracted media to be an important enough state interest to warrant such protection.
It's also possible that the court could view this as non-content-neutral, assuming it doesn't apply to tweets about things other than the ongoing event. Such a restriction would receive strict scrutiny, and likely be struck down because the state is conditioning privileges on a requirement that a reporter not engage in protected speech (which a tweet during a basketball game is).
I think the time-place-manner analysis is more likely, but I'd have to do some case research to be sure.