Home from a great trip! Thanks again everyone for all the suggestions. Here's three notable places I visited for a cigar:
Park Hyatt Hotel - New York Bar: Up on the 52nd floor overlooking Shinjuku and the rest of the city, an amazing place for a cigar and an excellent selection of fine Scotches and Bourbons. A tad on the pricey side, it's very upscale, and after 8pm they charge a cover (around 2000 Yen) for live music, usually light jazz. There's a super nice restaurant attached, the New York Grill.
Ace Cafe (Kyoto) - 10th floor bar overlooking the Gion District, just of Sanjo-dori overlooking the river. A little less refined, more of an edgy place for a hip, younger crowd. Cigar friendly but not enthusiastic about it, I wasn't sure if they were just being polite and accommodating or if they really allow cigar smoking. There's plenty of smoking in general going on so as usual why not cigars, right? Decent drink menu, for example I was able to have a moderately priced Yamazaki Distiller's Reserve here.
Twenty-Eight (Conrad Hotel Tokyo) - Great location in Shiodome, high up with a great view of the bay. Excellent drink menu featuring all the big names as well as several different Scotch flight offerings. Pricing was about what you'd expect from an upscale bar in an upper end hotel like that. Certainly cigar friendly with a few offerings of their own, although I didn't check cigar prices or what their stock was. Very glad to have been recommended this one, thanks mc808!