alternately

All night Katerina and I watched alternately, though, I shame to say, I slept more than I watched; and, having first ascertained that there was no change, I came directly hither.

Downstairs, he was informed. That is, down the escalators, where there is both a women’s room (often with a line) and a men’s room (almost never with a line). Or alternately, he could ask the station master to buzz him into the “family” bathroom. He opted to go downstairs.