This has certainly focused on the coffee places.

I would wholeheartedly agree with the recommendation for Turret coffee in Tsukiji area. They intermittently have a pop up shop in the Ginza area as well.

In the Shibuya area there is Chatei Hatou where they are very serious about pour overs. They will even do a pour over equivalent of espresso but it'll take 10-15 minutes. They do allow smoking in there as well.

The folks who used to do Omotesando Koffee have a newer place open, Koffee Mameya, but it is not a sit around with coffee sort of place. I think they are more geared to selling beans