Her father opened his shop in a premises further up Main Street on January 30, 1964. Soon a building became available a few doors down where Main Street meets Royal Parade and shares a corner with the bank.
"It was a funny coincidence that he moved in here," Eveleen explains, "while it had been a shop just before he bought it, prior to that, it was a dance hall, the Kincora Dance Hall. My father had his own dance band and, in the 1950s, often played here."
He swapped the stage for the counter and spent more than 50 years behind the till.
You could buy whatever you wanted in Jimmy's, from a litre of milk to a spool of thread, a ream of paper or a ride-on toy tractor.
As a news agency, the shop carried everything from the Financial Times to Ireland's Own. If the children needed school copy books or pencil cases etc, Jimmy's had it. Indeed, as a parent, you might get a text during the day asking you to go to Jimmy's for a geometry set, a packet of markers or a scrap book and drop them to the school.
The story of the shop is a veritable social history of the country.
"I can remember being in the room at the back bagging potatoes into half-stone and stone bags, and they were all locally grown," Eveleen recalls. "It's a good job the health and safety rules we have today weren't applied then.
"The big invention of the late '60s and early '70s was the whipped ice-cream machine. Ice cream was a huge seller, particularly on Sundays," Eveleen says.
Tea was also a huge thing. The van men from Lyons or Barry's would have a job to bring in the amount of tea leaves or tea bags needed every week. Nowadays, I could bring in the weekly supply under my arm."
The biggest change she noticed in recent years was in the toy market.
"My father used to go to the bigger toy stores once a year for toys; I had to go once a month to keep the toys stocked up."
However, too much changed too quickly and on the evening of Wednesday, March 6, the lower end of Main Street was packed as the locals gathered to say goodbye to Jimmy's.
There was a great sense of occasion and a deep sense of loss.
The loss is tangible; it's a personal thing. There are people who had orders for magazines and periodicals at Jimmy's going back years: fishing fanatics, motorcycle enthusiasts, crochet junkies and history buffs.
They were all known by name; their orders were anticipated, not just by the customers, but by Jimmy and the staff. If the publications didn't arrive on time, Eveleen or Marie, her loyal assistant, would be on the phone to the suppliers even before the customers came in.
Local children got summer jobs at Jimmy's and learned the retail trade from the best in the business. Whenever I missed the farming papers, they were kept for me or, if not, copies would be found.
About four years ago, a young French nephew of mine came to stay with me for a few weeks. The day before he left to go home, I went to Jimmy's to get a big Munster rugby flag for him as a souvenir.
When Eveleen heard why I was buying it, she put it under the counter and said: "Leave that with me. I'm expecting Anthony Foley to call, I'll get him to sign it." I collected the signed flag a few hours later and it now hangs proudly on a child's bedroom wall in Grenoble.
At Jimmy's, you were known, and the people you knew were known. You would be told quietly about a funeral or an illness or a bit of good fortune that you should know about.
This is the kind of connection that is lost, this vital thread in the social fabric of our community.
After the official closure, the steel shutters at Jimmy's remained down, but the door was open as the last of the stock was sold off. Customers arrived with bunches of flowers and stocked up on wrapping paper, diaries, yearbooks, writing paper, envelopes, notebooks and next year's Christmas cards.
Eveleen took the array of flowers to the graveyard where she laid them at the graves of customers gone on the last journey.
Jimmy now lives on the next street and, up to recently, would wait until the day's till receipts were brought to him before going to sleep.
Eveleen is keeping the news of the shop's closure from him, and when I mentioned I was thinking of writing something about it, she threw her eyes to heaven and said: "Another paper I'll have to hide from him."
The shutters are down and the door is locked. When I pass, my breath is taken away at the realisation that I will never go in to Jimmy's again.