you can walk to Ginza from the Tokyo Bay IC. A taxi ride from Roppongi to the hotel would be about 2,500yen (if you're going to Roppongi for 'action' you're unlikely to want to return before the last trains run... but there's a direct link from Hamamatsucho station to Roppongi via the Oedo line from Daimon station which is directly linked to Hamamatsucho - 3 stops)

A cab ride from the Strings to Roppongi would cost around the same. There's no direct train link although the journey is straightforward enough this way. When I checked out of the Strings to then check into the ANA IC I went by bus. The Toei 96 route runs from the Takanawa side of Shinagawa station to Roppongi http://www.kotsu.metro.tokyo.jp/bus/map/index.html
you can the see orange 96 route which goes to the North of Shinagawa 品川 here
http://www.kotsu.metro.tokyo.jp/bus/map/route/10_7.html
it then loops around at Roppongi before heading South again: http://www.kotsu.metro.tokyo.jp/bus/map/route/9_7.html
Easy and quick connection.
But don't be put off, Shinagawa is a very convenient location for those visiting Tokyo.