I'm a Philippine passport holder travelling to USA via Narita and, four hours later, Haneda.
Do I need a transit visa?
Yes, you will need a visa, as you'll be entering the country. The Philippines are not among the visa exempt countries for visiting Japan, and the Japan embassy in the Philippines has a list of all relevant documents / info that you may need.
Narita and Haneda are almost 50 miles apart, so you definitely have to exit the airport to get to the other.
We have a question on getting between the two airports.