Throughout Tokyo (and Japan) there are many, many small, independent coffee shops, normally run by one guy who's the owner and has been running the shop for decades. They often, but not always, have great coffee, often prepared in strange ways (siphon coffee for example) but are usually quite quirky. I've seen everything from old video arcade tables games used as tables to one dedicated entirely to James Brown. As the owners are one man shows, and often quite old, they do what they want and the decor, etc. reflects that. As many of them are quite old, they're slowly disappearing as their owners retire or die, and are generally not replaced so enjoy them while you can.
Smoking is mostly allowed (plus for me as a smoker) but don't let that put you off if you're not. Most have food of the coffee shop variety ( cheese toast, sandwiches, cakes, etc, some have lunch specials). Note: "oyagi" means something like "old, no-longer attractive, man". These are the kind of people who often go to these places, but it's not as bad as that sounds and these guys wouldn't go somewhere for years or decades if they weren't good.
Everywhere, just look for small signs advertising coffee (usually in English) on the street or shop windows. They're mostly on the ground floor.
Send your feedback or queries to firstname.lastname@example.org