"The Man Who Went to the Moon—Twice," a short story by Howard Rodman. One of the many stories which were first published in Dangerous Visions, a book edited by Harlan Ellison and released in 1967. I have just discovered that if I Google for the opening words of the story, I can go to a page on Google Books which allows us to start reading the entire text of the tale as a sample from Dangerous Visions.
The plot is very much as you described. Exact dates are not given, but a rural part of the USA in the early 20th Century feels about right for the first part, and the later part of the story happens roughly 80 years later than that. I agree with you that it feels like the sort of thing Ray Bradbury might have written.
Here are the first few paragraphs:
The first time Marshall Kiss went to the moon, he was nine years old,
and the trip was accidental. A captive balloon broke loose at the
county fair, and away it went, with Marshall in it.
It never came down until twelve hours later.
"Where've you been?" Marshall's pa asked.
"Up to the moon," Marshall answered.
"You don't say," Pa said, with his mouth slightly open and hanging.
And off he went to tell the neighbors.
As you recalled, he briefly becomes a local celebrity, with it being implied that most of the local community take his story at face value at the time. Then, we are told, as the years rolled past, Marshall grew up and got married, and after a while his wife had died, and his grown children lived elsewhere with their own spouses and children, and he was alone on the old family farm, and eventually was "pushing ninety pretty hard." (In other words, he must have been around 88 or 89; hence my statement that it had been about 80 years since his balloon trip.) He seems to have outlived everyone he had known in his youth; we are told he can walk down the street in the local town and not recognize any old familiar faces.
He finally makes a deliberate decision to stay out of town (i.e., on his own farm) for a solid month. He is wondering if anyone will bother to drop by to check up on him and make sure he's all right.
No one does.
He finally heads back into town and, as you said, tries to repeat that time when he was briefly a local celebrity by telling everyone that he's been way up to the moon and back. This gets little reaction from his adult listeners, and some appear skeptical. (Which angers him, even though the skepticism is richly justified.) A little boy is the sole exception who is impressed when Marshall claims he's been to the moon and back, twice, and the kid goes to fetch a friend of his (a little girl) so she can meet the man who's just back from the moon. By the time they arrive at the farm house, Marshall has died quietly, presumably of natural causes.
Then, in the final paragraph, the author gives us some further context regarding one of the reasons Marshall's claim was just shrugged off by the adults in the local community. This is obviously one of the bits you were paraphrasing from memory in your post.
That was the day the Mars rocket went on a regular three-a-day
schedule. Hardly anybody went to the moon at all, any more. There
wasn't enough to see.
It's worth reminding everyone that in the real world, when this story was written and then submitted to Harlan Ellison for his anthology, Neil Armstrong had not yet made that "one small step" onto the surface of the moon. That would occur in 1969. But Rodman was already anticipating a time when such trips would be old hat, instead of something new and exciting for a tourist to brag about having done. He discusses his own motives in an "Afterword" which I won't bother to quote or summarize. Anyone interested can read it via the Google Books link I provided.