"Marlon Brando's part in the movie was so convincingly Oriental, that many theater managers reported moviegoers coming to them and demanding their money back because (to put it simply) Marlon never appeared on screen."
The closest information I can find for a date is that Brando arrived in Tokyo on April 3rd 1956 and then headed to Kyoto for filming (and this candid) on April 8th. I can find no end date for the shoot. Initial release was on November 29th, 1956.