The Grand Hyatt is the only international chain hotel in Roppongi. There are no chain hotels in Shibuya that I know of. The Hilton is in Shinjuku (as are the Keio Plaza, Century Hyatt, and Park Hyatt), however there is a nearby subway line that goes to Roppongi in 11 minutes.

The Westin is in Ebisu, and there is a subway line that goes to Roppongi in 5 minutes.

From Ebisu or Shinjuku, you can take JR to Shibuya in 3 minutes or 8 minutes respectively.

Although Roppongi is fairly close to Shibuya (maybe 20-25 minute walk), there isn't really a direct connection via subway that I am aware of.

The Grand Hyatt is very close to the subway station. The other hotels are further from the subway station, but not too bad, from 3-8 minutes, except for the Westin which is about 10 minutes walk from both JR and the subway.

There are a zillion restaurants in Tokyo, aside from recommendations, you could get the Zagat Tokyo restaurant guide (but bear in mind that although it is excellently translated, the recommenders are all Japanese who are not all that familiar with Western cuisine).