The cocktails in Japan in general are highly over rated in general. There are a few absolute gems though.
Market fresh seasonal cocktails.
Location: 1-6-4 Azabu-Juban, Anniversary Building 1F, Minato-ku, Tokyo 106-0045
Bar High Five
Of the orignial cocktail bars from the "golden era" of Japanese bartending and cocktails; this is the only one still doing good work.
Location: 104-0061 Tokyo, Chuo Ginza 7-2-14, NO.26 Polestar Building, 4th Floor
Surprisingly good cocktails. Although the menu is rather short. Usually about 5 cocktails. They also have a remarkable spirits collection.
Location: 2 Chome-25-11 Nishiazabu, Minato City, Tokyo 106-0031
Bar Caol Ila
Decent cocktails if you come on the weekend, but the main reason to come is for scotch. Not just Caol Ila, but all scotch distilleries. They have a tremendous variety and I love drinking here. It is challenging to drink here if you do not speak Japanese, but I don't speak Japanese and I still adore it.
Location: 150-0043 Tokyo, Shibuya City, Dogenzaka, 1 Chome-13-3 MST 3F
Shot Bar Zoetrope
Probaly the best selection of Japanese whiskey in Tokyo. Be sure to try their flights as they are incredibly well priced and a great way of trying several things.
Location: 7 Chome-10-14 Nishishinjuku, 3rd floor, Shinjuku, Tokyo 160-0023