Will you have checked baggage? Because airlines will not transfer your checked baggage between NRT-HND. You have to carry your own baggage between NRT and HND.

If you have checked baggage you should head directly from Narita to Haneda, check your baggage at Haneda, and then head out to Tokyo for dinner. If you do not have checked baggage then you can head out to Tokyo directly from Narita.

I understand Narita Express train or Limousine Bus is pricier. However, Narita Airport is located 58 km (36 miles) from Tokyo. Arriving at Narita 3:30 p.m. and departing Haneda at midnight sounds like a lot of time. But as first timer not familiar with Tokyo, I will recommend Narita Express or Limousine bus because it will be easier to figure out and save time.