From Haneda, the Keikyu line or Monorail reach central Tokyo in about 30 minutes. From Narita allow 60 to 90 minutes via the N'EX or Skyliner.
Get a Suica or PASMO IC card and ride the JR and Metro lines everywhere. Trains stop around midnight, so plan your trip back before the encore.
Plan your route to Tokyo Dome ahead of time, and leave early to beat the crowds.