to catch the Shinkansen I have to take taxi to Tokyo Station No, you do not have to do so.
In Tokyo, Tokaido Shinkansen of JR-Central is available in Tokyo and Shinagawa Stations.
(Maybe you read something about a Shinkansen line of JR-East, which is available in Tokyo Station but not in Shinagawa Station, and thought as if it were your Shinkansen line.)
Hotel Metropolitan Marunouchi and Hotel Ryumeikan Tokyo are near Nihombashi Gate for Tokaido Shinkansen tracks of Tokyo Station.
Shinjuku Highway Bus Terminal are away from these two hotels.
From Hotel Metropolitan Marunouchi you can walk also to recently-restored Marunouchi Entrance of Tokyo Station.
From Hotel Ryumeikan Tokyo you can walk also to Nihombashi Station of Tokyo Metro, from which you can use Ginza Line to Ginza, Asakusa or Ueno.
Hyatt Regency Tokyo is near Shinjuku Highway Bus Terminal, where you will take a bus for Kawaguchiko.
Tochomae Station on Toei subway Oedo Line is close to this hotel.
Near this hotel are Tokyo Metropolitan Government Buildings Nos. 1 and 2.
TMG Government Building No. 1 has observatories at about 200 meters from the ground, which are open to the public free of charges.
A route from Hyatt Regency Tokyo to Kyoto Station:
Hyatt Regency Tokyo
--Walking--
Entrance A7: Tochomae Station (E-28): Track 1
| [Toei subway Oedo Line: Train, about 5 min., 180 yen]
Track 1: (E-26) Yoyogi Station: Exit A3
--Transfer--
North Gate: Yoyogi Station: Track 2
http://www.jreast.co.jp/e/stations/e1654.html| [JR Yamanote Linefs inner loop: Train, about 20 min.]
Track 1: Shinagawa Station: Shinkansen South Transfer Gate
http://english.jr-central.co.jp/info/station/_img/shinagawa_L.jpghttp://www.jreast.co.jp/e/stations/e1654.html| [JR Tokaido Shinkansen]
Kyoto Station.