Nora Barnacle was walking when she was approached by James Joyce on June 10, 1904.
She had just gotten off work at Finn's Hotel, where she was employed as a chambermaid. Six days later they would meet again (and snog some), and the fateful date for Ulysses (and Bloomsday) was set: June 16th.
It should be noted that Joyce was a bit indiscriminate in his attentions: He tried the same brazen approach the following day with another young woman -- but this other gal was accompanied by a male companion who beat the tar out of Joyce. The stranger, a fellow Dubliner who happened to be a Jew, helped Joyce clean up afterwards; he would become the model for Leopold Bloom.