Is it all right to use soap in a Japanese bathtub?

Usually, no. In Japan, people wash outside the tub using a showerhead or a wash bowl. This is why a typical Japanese bathroom has a drain in the floor. Once clean, people get in the tub and soak in the hot water. Since they are clean when they get in, everyone in the family shares the same bath water, eliminating the necessity to refill and to reheat new water for each bath (a Japanese bathtub is very deep, so it takes a lot of hot water to fill it up). In general, to keep the bath water clean you shouldn't use soap inside the bathtub in Japan.

However, in very small apartments, the bathtub and the toilet are sometimes placed in the same room to conserve space. If there is no room to wash outside the tub, no drain is built into the floor. In those instances, one would bathe in the tub as one does in America.

In short: drain on floor, no soap in tub. No drain on floor, soap in tub.