Back around the summer of 1972, a friend in the Navy told a time travel story to group of us. It was about an archeologist working in the Gobi Desert of China. He found a cave made into a makeshift jail with the ancient remains of a man in it. Scribbled on the back wall of the cave was modern Japanese writing that said that the man had been blasted to the past from Hiroshima in 1945. I have been trying ever since to find evidence that this story had actually been written. I thought the internet might help but not so far. If anyone who reads this happens to know anything about this story, I would be very grateful for your help.
Michael Moorcock's Warlord of the Air also features a man thrown into the past (which was really an alternate past from an alternate future to his real past...) as a result of an atomic bombing (but not the atomic bombing) of Hiroshima as a major plot point, as well as the same man travelling in time on another occasion as a result of an incident in a cave. It was originally published in 1971, so it fits the time period you're remembering. Perhaps your friend took that idea and embellished it a little?
The core plot feature (Hiroshima victim sent to past) sounds a bit like the 1948 short story "The Monster" by Gerald Kersh, but many details are different. Since it sounds like you heard this story second-hand, it's possible the teller confused/modified it?
Like some Random Passerby said, the story might have changed through the arab telephone.
It makes me think of a film. Well, two films, which are mixed in my head.
One is “the Philadelphia Experiment”. During the Second World War, two young american soldiers get zigzapped 30 years or so into the future.
The other film is the reverse story. It is named “the Final Countdown”. A big modern war ship, an aircraft carrier, gets zipzapped in the battle of Pearl Harbour. The japanese have a hard time.