I don't know how things work in Tokyo, but in Osaka there were basically three ways a small business like that survived.
1) Most commonly, the shop owner actually owns the building and lives above the bar. I think this goes for most small businesses that seemingly have no customers and it's hard to imagine they turn much of a profit, like small izakayas, futon shops, soba joints, karaoke bars, etc. Seems like this concept is dying out with the older generation, as most kids these days don't aspire to live in their parents' home forever and run the inherited ramen shop for another fifty years. But there are upsides to this--a friend of mine married into money and his wife's family owned large parcels of both land and the buildings on them (usually it's one or the other) right next to the train station, and he rented out space to two large chain izakayas for exorbitant amounts of money and used the profits to open his own little bar upstairs which made zero money and was basically playground for him and his friends (including me) where we would drink and gamble all night.
2) The shop is small and the space rented but has a dedicated group of regulars who spend a dedicated amount of money there. This is like my regular bar for all the years I lived in Osaka. Small counter bar, perhaps a dozen seats, but every seat was occupied every night from 5pm to 1am with regular customers who generally had good jobs and money and each person, including all the drinks they'd buy the owner/bartender, would drop at least 10k a night and probably more on weekends. Nobody got rich but you could eke out a living on that.
3) It's a yak money laundering front.