There is a package where you get round trip bus to/from HND + Tokyo Metro pass:
https://www.limousinebus.co.jp/en/gu...ubwaypass.html (note that Tokyo has multiple Metro providers so this is only valid on the Tokyo Metro - it should be sufficient)
When I went, the Shinjuku Hilton was the last / first stop on the route. The bus is very good - they do all the lifting, wifi and bathroom onboard. I bought the package and then asked the hotel to book the return trip.
The underground tunnel connects you to different Metro stations. They also have a little shuttle that takes you to Shinjuku (they are in West Shinjuku).
Also check out the free observatory in the Tokyo Metro Government highrise. Windows might be dirty from people's hands but it is free
