I'm trying to think of the short story about a condemned prisoner that experiences a lifetime between the time a bullet is fired and the time it ends his life.
I feel like the story might be Russian? Or maybe Borges?
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"Jaromir Hladik, a Czechoslovakian writer, confronts a German firing squad. He prays to God for a full year in order to complete his play. A tear starts from his eye. In the time that the tear falls, he mentally finishes, without hurry, revising as he writes, his play. When the tear finishes its fall, at that moment, he is shot to death by the firing squad."
The story we're looking for here is almost certainly the Borges story suggested by Dragan Milosevic. Still, this other story, by a much less famous writer, does more or less match the terse description given in the question; so I've decided to leave this answer here, in case someday someone who is looking for that obscure writer's story runs across this question.
"Moment Without Time", a novelette by Joel Townsley Rogers, previously identified on this site as part of the (unaccepted) answer to this question; first published in Thrilling Wonder Stories, April 1952, available at the Internet Archive. A story by an American writer in an American magazine, but with a Russian setting. Here is the editorial blurb from the original magazine publication:
Time stopped for Piridov, deferring his death-sentence while the bullets
were yet in midair, and in that endless—
MOMENT without TIME
—he became the most dangerous man in Moscow!
The dissident physicist is about to be executed, when he suddenly hits on the solution to his research problem:
Piridov was facing the firing squad in the execution cellars when the solution hit him.
It was the answer he needed, the answer to the mathematical relation of time to the three spatial dimensions. It came to him in the form of an equation, as simple as the German's historic energy-mass formula—"e=mc2", where "c" represents the speed of light.
[. . . .]
Sergeant Death had shouted "Fire!", it seemed to Piridov, minutes ago. The waiting was painful. Slowly and reluctantly, he opened his eyes on the cellar scene.
The white-painted walls lit by thousand-watt bulbs, the four rifle muzzles pointed at him, the squinted eyes behind the sights, the taut retracted trigger-fingers, squat thick-chested Sergeant Death standing to one side with his hand down at his thigh in the conclusion of the sweeping axe-blade gesture with which he had accompanied the command to fire, the bullets coming at his breast. Four of them, straight at his heart, not more than twelve to fifteen inches away; while from the corner, out of the range of fire, the one-eyed scavenger directed the stream of water around his feet to wash the blood away.
No, the bullets weren't coming. They were motionless in space. The eyes behind the sights were motionless. The water hosing over his feet was motionless, both the stream and the splattering drops of it. The red second hand of the electric clock on the wall above the steel door was motionless, the minute and hour hands straight up and down at six.
Piridov spent not a lifetime but some hours in the "Moment Without Time", enough to walk around Moscow and visit various people and places including his wife Anna. Finally, he returns to his unfinished execution:
He walked the long miles back, with bowed head and limping gait, toward the city's center and the great prison. He went in, closing the entrance doors behind him. He went through the other doors, closing each in turn, along the corridors, down the flights of stairs. At the end of the long brightly-lit corridor in the deepest basement he went towards that farthest door. He went back in the cellar room, closing the door behind him.
The noseless fire-burned Sergeant Death stood with his hand down-moving. The riflemen stood with their fingers squeezed on their triggers. The one-eyed janitor hosed down the floor in motionless stream and spatter.
Piridov went past them as quietly as a shadow. At the farthest wall he turned, facing the squad, placing his feet carefully on the precise dry spots, in the empty spaces of the spattering water, which marked where he had been standing. The bullets in their swift rifled flight were poised twelve inches from his breast.
Piridov closed his eyes.
"I had a dream," he told himself. "A dream of a bright equation within a timeless instant. Of things I love and things I hate. Of a sword sweeping. A dream of water. But I am very tired. So let the dream be over."
And it was . . .
The rest of the story tells what happened with the various messages he left around town during his timeless moment.
Most probably Dragan Milosevic's answer is the one you're looking for. But since you say "the story might be Russian", I wonder if you're also conflating it with elements of Dostoyevsky's The Idiot, in which execution is an important theme and one based on Dostoyevsky's own experience. The topic of condemned prisoners at the moment just before their execution is discussed several times in the novel. From Wikipedia (emphasis mine):
In 1849, Dostoevsky was sentenced to execution by firing squad for his part in the activities of the Petrashevsky Circle. Shortly after the period of interrogation and trial, he and his fellow prisoners were taken, without warning, to Semyonovsky Square where the sentence of death was read out over them. The first three prisoners were tied to stakes facing the firing squad: Dostoevsky was among the next in line. Just as the first shots were about to be fired, a message arrived from the Tsar commuting the sentences to hard labor in Siberia.
The experience had a profound effect on Dostoevsky, and in Part 1 of The Idiot (written twenty years after the event) the character of Prince Myshkin repeatedly speaks in depth on the subject of capital punishment. On one occasion, conversing with the Epanchin women, he recounts an anecdote that exactly mirrors Dostoevsky's own experience. A man of 27, who had committed a political offence, was taken to the scaffold with his comrades, where a death sentence by firing squad was read out to them. Twenty minutes later, with all the preparations for the execution having been completed, they were unexpectedly reprieved, but for those twenty minutes the man lived with the complete certainty that he was soon to face sudden death. The Prince recounts in detail what the man experienced during those twenty minutes.
The subject of capital punishment first comes up earlier in Part 1, when the Prince is waiting with a servant for General Epanchin to appear. Engaging the servant in conversation, the Prince tells the harrowing story of an execution by guillotine that he recently witnessed in France. He concludes the description with his own reflections on the horror of death by execution:
... the worst, most violent pain lies not in injuries, but in the fact that you know for certain that within the space of an hour, then ten minutes, then half a minute, then now, right at this moment – your soul will fly out of your body, and you'll no longer be a human being, and that this is certain; the main thing is that it is certain. When you put your head right under the guillotine and hear it sliding above your head, it's that quarter of a second that's most terrible of all... Who can say that human nature is able to endure such a thing without going mad? Why such mockery – ugly, superfluous, futile? Perhaps the man exists to whom his sentence has been read out, has been allowed to suffer, and then been told: "off you go, you've been pardoned". A man like that could tell us perhaps. Such suffering and terror were what Christ spoke of. No, a human being should not be treated like that!
Later, when he is conversing with the Epanchin sisters, the Prince suggests to Adelaida, who has asked him for a subject to paint, that she paint the face of a condemned man a minute before the guillotine falls. He carefully explains his reasons for the suggestion, enters in to the emotions and thoughts of the condemned man, and describes in meticulous detail what the painting should depict. In part 2, the usually comical character of Lebedyev also considers the horror of the moment before execution. In the midst of a heated exchange with his nihilist nephew he expresses deep compassion for the soul of the Countess du Barry, who died in terror on the guillotine after pleading for her life with the executioner.