**More spoilers**
I agree that there were some problems, like the police forgetting about the cops that are sent to her home at the beginning, her sister's death not being too much of a big deal, and all the TV journalists broadcasting it all.
However, the obsessed fan doesn't just "happen to be hanging around the studio". He was there because he was invited to since it was her very last show, and the other guest (I think it was going to be the new DJ, who had been murdered by the psychopath) couldn't make it.
He actually was autistic or had some other kind of mental disorder. You can clearly tell it from his behavior throughout the film. He already was likely obsessed with her long before the radio show even started (remember that she was a TV reporter years ago), so it's not too much to imagine that he listened to her first episode and remembered the first song that she played. The killer also knew it because he recorded all or many of the episodes.
The only other thing I think she asks the fan is what she had said about Taxi Driver, which was also a memorable night because of the way that day's caller (as it's revealed later in the movie, the psychopath himself) acted on the phone.
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