It's easy to find Dům Shani as it's located facing the statue of the Rod of Aesculapius, on Trh ve Chrámové čtvrti. The building is owned by an old lady with a difficult personality; one has to curry favor with her to get upstairs. It is also home to Alvin in Kapitola III if Geralt chooses Shani as the boy's guardian.
Whenever Shani is at home, and of course, if Geralt and Shani are on speaking terms, he can stay for free. Shani hangs out there before and after work during Kapitola II, and in Kapitola III she seems to be permanently in residence, or not speaking to the witcher.