It is important to remember that in "A Dangerous Game," Rainsford is running for his life from General Zaroff who thinks it is great sport to hunt human beings like animals. The short story helps portray the irony concerning the way that hunters feel it is OK to treat the prey they are hunting. It captures the irony through the fact that General Zaroff, in the absence of having live animals to hunt on the deserted island, feels it is OK to hunt any humans that find their way onto the island instead. In the beginning of the story, General Zaroff and Rainsford are paralleled in their beliefs that the strong have the right to use the weak for their pleasure, as we see in Zaroff's own line, "The weak of the world were put here to give the strong pleasure." Since animals are weaker than mankind, Rainsford enters the story believing that mankind has the right to hunt and use animals for mankind's pleasure. Likewise, Zaroff believes those who become stranded on the island are the "scum of the earth" and that he is therefore justified in using them for his own personal pleasure. However, Rainsford soon learns that even the weaker have feelings that need to be respected when he himself begins to be hunted by General Zaroff.
At the moment that the narrator says of Rainsford, "He lived a year in a minute," Rainsford is crouching by a tiger pit he has dug up and planted with sharp stakes to kill anyone or thing that falls into the pit. His hope is that Generald Zaroff will fall into the pit and that Rainsford's life will be saved, and Rainsford is eagerly waiting for this event to take place. It also helps to remember that waiting is an agonizing process, especially if you are waiting to see if you will die or be spared. As he waits for Zaroff to pursue him, hoping he'll fall into the hidden pit, Rainsford feels like every minute that passes is dragging by because each minute is of vital importance to him. Hence, as he crouches waiting for what feels like an eternity, he feels like the length and actions of an entire year can fit into one minute in which he waits for Zaroff to fall into the pit.
Hence, what Connell means by describing a year in one minute is simply that waiting for either your death or your salvation to take place is an agonizing process, so agonizing it feels like all the events that could take place in a year are actually taking place in a minute, dragging the minute on and on.
When Sanger Rainsford realizes that his host, General Zaroff, has begun hunting humans, Rainsford is appalled and disgusted. When Rainsford discovers that he is going to be the general's next prey, he is frightened, and that is not surprising.
Of course, Rainsford is a world-class hunter and an equal match for Zaroff, who is also a superior hunter of big game, so Rainsford assumes he will be able to outmaneuver the general and save his own life. What he discovers on the first night of the hunt, however, is that his assumption that his careful and intricate maneuvers would outwit Zaroff is wrong. The general is able to track his prey, even at night, and Rainsford is afraid he will be killed. Zaroff is intrigued enough by this hunt to let Rainsford live--for now.
By the next day of the hunt, however, Rainsford is using every trick he knows to stay alive, and he hopes they will be successful. After he makes the Burmese Tiger pit, Rainsford waits to see if his plan has worked and whether Zaroff will be caught in the trap.
He knew his pursuer was coming; he heard the padding sound of feet on the soft earth, and the night breeze brought him the perfume of the general's cigarette. It seemed to Rainsford that the general was coming with unusual swiftness; he was not feeling his way along, foot by foot. Rainsford, crouching there, could not see the general, nor could he see the pit. He lived a year in a minute.
What the expression "lived a year in a minute" means is that in that short, tense moment, Rainsford is so afraid that he feels as if he lives an entire year in that one moment. The expression is generally used by those who are under great fear and/or distress which is strong enough to cause this kind of reaction.