Why is Shukhov's spoon a significant symbol in One Day in the Life of Ivan Denisovich?

Shukhov's spoon is an important symbol because its symbolism (1) adds to his characterization and (2) reveals how he survives prison camp. [It might be noted that "Shukhov" is another name for Ivan Denisovich: "Shukhov dashed in ... 'Didn't land in the hole, then, Ivan Denisovich?'"]The spoon symbolizes survival and human dignity.

Two symbolic representations related to the spoon build characterization: (1) that he made it himself and (2) that he remembers to bring it with him even in unexpected situations. The first develops Shukhov as an ingenious, industrious man, who does not yield passively to his situation: he is active in bettering it, even if in only the smallest degree. The second develops him as quick thinking, alert and conscious of self-preservation: taking care of minute details can mean the difference between survival and death.

[Shukhov] took the bucket, and just as he was, without mittens (he'd left them under the pillow in the rush), went out to the well.... [He'd] tossed his foot rags ... (his spoon tinkled as it hit the floor — he'd had to get ready for the hole in a hurry, but he still hadn't forgotten his spoon)....

The spoon symbolism reveals how Shukhov survives in prison camp. Shukhov carries his spoon everywhere he goes, even if suddenly call to some other task. For instance, when he lies in bed too long on the morning the story opens and is punished by Ivan-and-a-half, he forgets his mittens under his pillow thus is forced to draw water from a frozen well with a frozen rope, yet he does remember to put his spoon in his felt boot. He knows what is needed to survive. Then, when he eats his skilly and porridge with his spoon, Shukhov insists on removing his cap. This is a sign of recognition of his dignity and ultimate importance. Thus the spoon symbols reveal that he survived through remembering the foremost needs of life and the foremost value of life: nourishment and dignity.

Next, he removed his cap from his shaven head — however cold it was, he wouldn't let himself eat with his cap on

What is the significance and impact of the setting and the spoon symbol in One Day in the Life of Ivan Denisovich?

Usually "impact" refers to dramatic impact of a play or literary element on the audience or reader. For example, the impact may be to produce suspense or sympathy or immediacy. When you combine "impact" with "significance," though, this refers to literary importance, like the meaning of symbolism, the affect of setting on mood, etc: it refers to things that are part of critical analysis (also called literary analysis or critical appreciation in various cultures).

First, we'll address your broader question of "How do I find the significance and impact?" Starting with setting, you look for how the setting, which is time and place, affects the mood of the story or foreshadows the resolution of the story or reinforces the characterization of the character(s) of the story. For instance, is the setting under a blazing sky on a storm-tossed ocean with men struggling for survival in a small boat, like in Crane's "The Open Boat?" If so, the setting impacts the mood, the foreshadowing of the resolution and the characterization of the men struggling against the sea since part of the "place" portion of the setting would be the order or disorder within the boat.

Finding the significance and impact of symbolism may be a bit more difficult since symbolism is meant to communicate through an underlying figurative, non-literal message. To find the significance and impact of symbolism, you must first recognize the symbols. Symbolic objects or ideas may be attached to characters, places, objects, or abstract concepts. For instance, an emerald pendant can symbolize a character's loyalty and devotion. A wall, an object, can symbolize the idea of racial segregation as in Hughes' poem "As I Grew Older." Or the action of giving can symbolize the abstract concept of altruism as in Lowry's "The Giver."

The setting in Solzhenitsyn's One Day in the Life has the literary impact of being integral to plot development. It is the stark cold and barren wasteland they are in that complicates and underpins all conflicts that arise:

You could count on a month with nowhere to go for a warm, ....  You wouldn't even be able to light a fire out in the open ... Your only hope would be to dig, dig, dig, for all you were worth.  

While the setting also adds to mood (atmosphere)--though most of the mood is set by the tone of the narrator--it also contributes to characterization while we see how they respond to the stark conditions and cold.

The spoons they each have symbolize both survival and civilization. One must eat to survive. It is with one's spoon that one eats. The spoon is a strong symbol of survival.

First he just drank the juice [broth], spoon after spoon.

It symbolizes civilization (1) because with a spoon, you are not reduced to eating with fingers and (2) because with your spoon, you have a few minutes three times a day to slowly, leisurely feel pleasure and satisfaction, to feel human:

He started eating slowly, savoring it, just the same.  If the roof burst into flames, he still wouldn't hurry.