1. There are 02:00 hour late night buses departing from Haneda Airport International Terminal to various places within Central Tokyo (Ginza, Tokyo Station, Akihabara, Asakusa, Shinjuku, Higashi-Shinjuku, Ikebukuro): www.limousinebus.co.jp/guide/accessbus.html
See pages 40 and 43 on this E-booklet inside the circles: limousinebus.co.jp/en/img/timetable_h_en.pdf
2. Book a hotel near the airport. There are plenty of hotels in Anamori Inari area, like Hotel JAL City, Hotel Mystays Haneda, Toyoko Inn, etc. You might expand your search to Kamata which has dozens of hotels.
3. Find an empty seat and sleep for a while until the first trains and bus (many people on flights arriving at awkward hours do this; Haneda Airport could be the safest place on earth to do this although you'd be watched over by guards and cameras).
4. There is at least a currency exchange, shower room, a couple of cafes open throughout the night, if you don't feel like taking the bus, don't want to book a hotel, don't want to take a nap at the airport.
5. You can always take a taxi to your desired hotel within Central Tokyo - and that might make the driver quite happy, but it could end up being prohibitively expensive.Edited: 05 August 2018, 14:48
Could you tell how much are we talking when you say prohibitively expensive?