Short story involving time travel through a train station

I cannot remember the story exactly, but I guess it goes like this -

Protagonist was travelling somewhere. So he goes to a train station. But the ticket counter he went to appeared old fashioned (may be with old lamps and kind of things). He gives money to the man at the counter to buy a ticket. But seeing the coins, the person asks if the protagonist is trying to fool him. (Implies that the person has not seen the coins before, so he thinks protagonist is trying to fool him giving fake money).

Somehow the protagonist reached back to his home and tells his relatives/neighbors about his experience. He got curious about this and made several attempts to get back to that same ticket counter, but couldn't find it.

After some days the protagonist go missing. The relatives/friends find an old bundle of letters at his house. The letters had been sent by the protagonist from a different time period. From the letters it is understood that the guy had found the counter and traveled back in time. He is happy there and do not wish to return.

I remember the place of the plot as Illinois.

Answer

Short story involving time travel through a train station

"The Third Level" is a short story by Jack Finney, first published Collier's, October 7, 1950; reprinted in The Magazine of Fantasy and Science Fiction, October 1952, which is available at the Internet Archive. You can also read it here, or you can listen to actor Jonathan Frid read the first page or two in this YouTube video.

It starts like this:

The presidents of the New York Central and the New York, New Haven and Hartford railroads will swear on a stack of timetables that there are only two. But I say there are three, because I've been on the third level at Grand Central Station.

Protagonist was travelling somewhere. So he goes to a train station. But the ticket counter he went to appeared old fashioned (may be with old lamps and kind of things).

There were brass spittoons on the floor, and across the station a glint of light caught my eye; a man was pulling a gold watch from his vest pocket. He snapped open the cover, glanced at his watch, and frowned. He wore a dirty hat, a black four-button suit with tiny lapels, and he had a big, black, handle-bar mustache. Then I looked around and saw that everyone in the station was dressed like 1890-something; I never saw so many beards, sideburns and fancy mustaches in my life. A woman walked in through the train gate; she wore a dress with leg-of-mutton sleeves and skirts to the top of her high-buttoned shoes. Back of her, out on the tracks, I caught glimpse of a locomotive, a very small Currier & Ives locomotive with a funnel-shaped stack. And then I knew.

He gives money to the man at the counter to buy a ticket. But seeing the coins, the person asks if the protagonist is trying to fool him. (Implies that the person has not seen the coins before, so he thinks protagonist is trying to fool him giving fake money).

The clerk figured the fare — he glanced at my fancy hatband, but he figured the fare — and I had enough for two coach tickets, one way. But when I counted out the money and looked up, the clerk was staring at me. He nodded at the bills. "That ain't money, mister," he said, "and if you're trying to skin me you won't get very far," and he glanced at the cash drawer beside him. Of course the money was old-style bills, half again as big as the money we use nowadays, and different-looking. I turned away and got out fast. There's nothing nice about jail, even in 1894.

Somehow the protagonist reached back to his home and tells his relatives/neighbors about his experience. He got curious about this and made several attempts to get back to that same ticket counter, but couldn't find it.

And that was that. I left the same way I came, I suppose. Next day, during lunch hour, I drew $300 out of the bank, nearly all we had, and bought old-style currency (that really worried my psychiatrist friend). You can buy old money at almost any coin dealer's, but you have to pay a premium. My $300 bought less than $200 in old-style bills, but I didn't care; eggs were thirteen cents a dozen in 1894.

But I've never again found the corridor that leads to the third level at Grand Central Station, although I've tried often enough.

After some days the protagonist go missing.

Not the protagonist, a friend of his:

Louisa was pretty worried when I told her all this, and didn't want me to look for the third level any more, and after a while I stopped; I went back to my stamps. But now we're both looking, every week end, because now we have proof that the third level is still there. My friend Sam Weiner disappeared! Nobody knew where, but I sort of suspected because Sam's a city boy, and I used to tell him about Galesburg — I went to school there — and he always said he liked the sound of the place. And that's where he is, all right. In 1894.

The relatives/friends find an old bundle of letters at his house. The letters had been sent by the protagonist from a different time period. . . . I remember the place of the plot as Illinois.

Actually the protagonist/narrator, who collects stamps, finds among his first-day covers a letter from his friend Sam, writing from Galesburg, Illinois, in 1894.

From the Galesburg, Illinois Wikipedia page:

Writer Jack Finney, author of The Body Snatchers, uses Galesburg as a setting for several of his time-travel tales.