IF you would go out, it is worth going to central area, if you are to shorten your sleeping hours and sure you can wake up
Rather than monorails, take Keikyu line directly to Shinagawa, which is closed to Sheraton Miyako. However, you would need a bit further to get to busy area, such as Roppongi, Shibuya, Shinjuku or Gotanda from there.
Enjoy your night.
Or spend a night in a nothing-special hotel at the airport