Whats a “Japanese napkin,” in historical context?
Some googling turned up this link to a picture of a souvenir napkin from 1883, which would have been the type of thing O Henry might be referring to.
Don’t know about Japanese napkin, but as for your paper napkin question, paper started being introduced in the US as toiletry/kitchen items around the turn of the century, with Scott Paper company making paper napkins around 1930s. See this about toilet paper, and paper towels.