Get past a law can (and I think here does) mean to avoid the applicability of a law. For example, if the law dictates that that one may not drive in a certain lane if alone (but with a non-driving passenger it's okay), putting a passenger in the car solely for the purpose of being able to drive in that lane can be called getting past the law. Press law here seems to mean a law that affects the press, meaning journalists and the like. I suspect therefore that the sentence means: Homayoon, himself an eminent journalist, had tried to evade the applicability of a law imposed on members of the press that required them to balance their freedom with responsibility—in vain, as events evolved.