Welcome again to my country!
A 7-day Japan Rail Pass cannot pay off for this itinerary.
Where are your Tokyo and Osaka accommodations?
Day 4: leaving Tokyo to Hakone For your Hakone trip, 2-day Hakone Freepass (version designed to be used beyond Odawara) (sold at 4000 yen) is worth considering.
You can purchase this pass at a ticket vending machine in Odawara Station of Odakyu Electric Railway when you make a transfer to Hakone Tozan Railway (Odawara -- Hakone-Yumoto).
- Japan Guide: How to travel between Hakone and Tokyo
http://www.japan-guide.com/e/e5206.html- Japan Guide: Hakone Freepass
http://www.japan-guide.com/e/e2358_008.html- Odakyu: How to purchase tickets
http://www.odakyu.jp/english/traffic/howto/ticket/#free: Check "How to purchase a Freepass" on this page.
Until Odawara, you can use either
the Odakyu Odawara Line (Shinjuku -- Odawara),
the JR Tokaido Line (for the section operated by JR-East) or
the JR Tokaido Shinkansen (operated by JR-Central) .
Among the three, the JR Tokaido Shinkansen is fastest.
(Shinkansen trains which make a stop at Odawara are not very frequently available.)
Compared to Tokyo Station, Shinagawa Station of JR is much simpler, therefore sounds better for you to take a Tokaido Shinkansen train from.
- JR-Central: Map of Shinagawa Station
http://english.jr-central.co.jp/info/station/_img/shinagawa_L.jpg- JR-Central: Map of Odawara Station
http://english.jr-central.co.jp/info/station/_img/odawara_L.jpgWhen you choose the Odakyu Line, you might hope to take a Romancecar.
A Romancecar is a Limited Express train with higher class seats, which need reservation.
Shinjuku Station of Odakyu is in the huge and complicated Shinjuku station complex; I recommend that you in advance check your route to the platform.
Instead of Shinjuku Station, you can take a train from a station on the Chiyoda Line.
The Tokyo Metro Chiyoda Line and the Odakyu Odawara Line are jointed in Yoyogi-Uehara Station.
(Romancecar trains skip this station.)
Romancecar Metro Hakone 21 is scheduled to start from Kita-senju Station (C-18) at 9:49 and arrive to (OH-47) Odawara Station at 11:52, which makes stops at Otemachi (C-11), Kasumigaseki (C-08) and Omote-sando (C-04) Stations but skips Shinjuku (OH-01) and Yoyogi-Uehara (OH-05) Stations.
- Tokyo Subway Route Map
http://www.tokyometro.jp/en/subwaymap/pdf/rosen_en_1702.pdf- Odakyu Line Route Map
http://www.odakyu.jp/english/traffic/railmap/