Hakata to Narita Airport

Hi everyone,

We want to spend 4 days in Tokyo before we fly back home and I was wondering if a rail pass was the best idea.
We would be leaving from Hakata to Narita Airport and then the next day take the NEX to the city and back. We would use the NEX again there and back the day after as well.
I would like to know which train lines I can use and if the total cost of them all will make a pass worth it.
Is there a train line from Hakata to Narita? Price?
I know you can use the NEX with a pass so that is ~4800 yen each way, correct?
Can I also use the pass on trains within Tokyo so I can visit Shibuya, Shinjuku, Ikebukuro, etc? What would the approximate price be to travel to multiple stations in a day?
And finally, can I use the pass to travel from Hakata to Beppu? and price?

The other option would be a cheap JetStar flight at around 8700 yen, NEX (4 times), and the train around the city after we get in.

Thank you for your help!