If you’re committed to doing a bit of driving the night you get in and spending a day somewhere on the way to Cape May, you could look into Princeton or Asbury Park. Princeton’s not a beach town but it has a nice downtown area, parks, restaurants, cafes etc.

That said, Cape May is beautiful and I would echo everyone else who suggests just crashing near the airport and getting a start first thing in the morning. Maybe stop at Princeton for breakfast (or lunch depending on how late you get on the road)