The chain coffee shops have largely taken over in Tokyo, not only Starbucks, but also Doutor, Tully's, and places that try to look like Starbucks and come just short of trademark infringement.
Your best bet for finding old-style local coffeeshops, the ones where the owner chooses the crockery, the decor, and the type of music played on the sound system, and serves toast, coffee, salad, and a hard-boiled egg before 10AM as "morning service," is looking on the side streets or heading toward the less touristy neighborhoods.