The restaurant in the series is supposedly set in Shinjuku's Golden Gai, but there doesn't really appear to be anything quite like it there.
However, this article (in Japanese) asserts that the actual inspiration was a real kushikatsu (fried stuff on a stick) joint called Yagura Kushikatsu in Tennoji, Osaka:
The connection is a bit convoluted though. Like the series, Yagura's regulars include taxi drivers, nightlife workers, and others up in the middle of the night. One of them was famous rakugo actor/singer Jakusaburō Katsura, who wrote a song about the restaurant called Yagura March (やぐら行進曲) in 1996, talking about a magical shop that opens at midnight with characters including those taxi drivers and a "Cinderella forgotten by time". Japanese manga artist Yaro Abe heard the song, liked the concept, and that's how the Midnight Diner comic and later TV series were born.
However, Abe never visited the restaurant, so many details like the idea of not having a menu are not the same. Also, the original Yagura relocated from the original Kyobashi to Tennoji some time ago, and the new location is open during regular business hours, not in the middle of the night. Perhaps easier for tourists, but not quite so atmospheric.
For what it's worth, in my student days I used to be a regular at a very Midnight Dinerish establishment in Shimokitazawa, Tokyo, which was open very late and where the menu consisted of whatever was on hand that day (the deep-fried river shrimp/kawaebi were to die for). However, the owner had a drinking problem, ran into trouble with some gentlemen in loud suits missing their pinky fingers, and had to leave town on short notice. Hope you're doing OK, Kashihara-san...