Rashomon is one of Akira Kurosawa's movies - I haven't seen it, but was vaguely familiar with the name. The short story doesn't appear to be much like the movie. Harrison Public Library has, for some reason, a collection of stories by Akutagawa that appears to have been originally published in Japan - all the notes, copyright information, etc. is in Japanese, and it is a side-by-side translation with English on one side and Japanese on the other. No Amazon quicklink for this one.
Accordingly, I don't know anything about these stories other than what I can gather from internal information. My guess is early twentieth century, although there is a fairy-tale quality that suggests the author drew heavily on traditional stories.
Once again, I don't have the cultural context to correctly interpret some of these stories. But they are beautiful, multilayered puzzles to ponder over. I most enjoyed 'The Handkerchief', in which a professor encounters an expression of human emotion, and 'The Spider's Thread' in which the Buddha attempts to rescue a soul from Hell. The imagery Akutagawa uses is transfixing.