As Snopes says, this is a legend with no photographic evidence behind it, and no specific place or year when it was seen. It is almost certainly false, and I agree with their logic:
... the mixing of Christian crucifixion iconography and Santa Claus is an unlikely pairing, even to non-Christians. Nativity scenes, not crucifixes, are the religious displays featured at Christmastime, and anyone with the least bit of thoughtfulness would have to wonder why a smiling, happy, jolly figure would be depicted hanging from boards with nails driven through his hands and feet. Santa Claus in a creche might be a plausible mistake (there are claims that figures such as the Seven Dwarfs have been spotted standing in for the Three Wise Men in various parts of the world), but a crucified Santa challenges credulity. As parody it's believable; as an honest mistake we find it implausible.
Furthermore, Santa and Christmas were already well-known in Japan due to foreign literature like Dickens, and the persistent and futile efforts of missionaries to convert the locals. Christmas was a national holiday from 1927-1947 since it happened to coincide with the death of Emperor Taisho in 1926. The idea that by the postwar period, when this supposedly happened, there were still designers in Japan unfamiliar with Santa seems terribly unlikely.
Here's Japanese Santa doing his job in 1916:
How did he get in without a chimney? That's a mystery no skeptic can solve.
Postscript: When you Google these keywords, you will more likely than not see this photo:
This is a photomontage made by a Japanese artist specifically attempting to recreate the legend for artistic effect. Here's the original source (note that clicking "Next" goes to a NSFW image).