Batman stops a young woman, Viola Vane, from killing herself. It seems that she had come to Gotham to be a great actress, but had only succeeded in becoming an understudy. However, she'd been writing her parents that she had become a famous star. Her parents were to visit, and Viola could not stand the shame.
Batman offers to help her. He goes around the city telling others her story, getting them to agree to help her out, and then, as Bruce, he escorts her and her parents around town. After an altercation with some thieves who stole the furs and jewels Viola was borrowing, he escorts her parents to see a play, who lead actress has agreed to allow Viola to play the lead part in her place. She does so well, that it's agreed she will take the lead permanently, when the current lead has left for Hollywood.
Although the 'State Building' has been mentioned before, I think this is the first time it's been clear that it must be the Empire State Building--it's described as having 102 floors. I don't believe it's been so explicit that Gotham City is indeed New York City, before this.