He stuck in his fingers, and pulled out—a nose! [...] His hands dropped to his sides for a moment. Then he rubbed his eyes hard. Then again he probed the thing. A nose! Sure enough a nose! Yes, and one familiar to him, somehow! Oh, horror spread upon his face! (1.7)
You have to love how Kovalev experiencing fear looking at a facial feature… and that fear is being reflected on his own features. Like, we aren't told that he feels bad, but instead that the badness can be seen on his face. But, uh, without much a face left, how does that horror spread? You can practically hear Gogol laughing at us.
So [the barber] sat silent. At the thought that the police might find the nose at his place, and arrest him, he felt frantic. Yes, already he could see the red collar with the smart silver braiding—the sword! He shuddered from head to foot. (1.14)
Fear seems to pretty much be the barber's main emotion. He's stressed at the idea of exposure and being caught, which is fine, but, uh, shouldn't there be other feelings? Guilt or innocence? Confusion about how this nose came to be in his bread roll? Sudden and dramatic loss of appetite?
"Stop, Prascovia Osipovna! I'll wrap it in a rag, in some corner: leave it there for awhile, and afterwards I'll take it away." […] But at last he got out, and donned waistcoat and shoes, wrapped the nose in a rag, and departed amid Prascovia Osipovna's forcible abjurations.
His one idea was to rid himself of the nose, and return quietly home—to do so either by throwing the nose into the gutter in front of the gates or by just letting it drop anywhere. Yet, unfortunately, he kept meeting friends, and they kept saying to him: "Where are you off to?" or "Whom have you arranged to shave at this early hour?" until finding a suitable moment became impossible. (1.10-16)
The barber is totally unable to commit to a plan of action. Everything he does is totally reactive—to his wife, to the nose, to friends he meets on the street. Are we supposed to think that he's different from Kovalev—or is this something the two dudes have in common?
He approached a mirror in some trepidation, and peeped therein. Then he spat.
"The devil only knows what this vileness means!" he muttered. "If even there had been something to take the nose's place! But, as it is, there's nothing there at all."
He bit his lips with vexation, and hurried out of the restaurant. No; as he went along he must look at no one, and smile at no one. (2.5-11)
Ok, so… what exactly is making Kovalev scared here? He's holding up the hanky, so he's clearly stressed about being seen—but is he assuming the reaction of other people will be disgust? Terror? Not really, right? And that's what makes this funny—he's basically acting as if he has a giant zit on his face. His main fear is that he'll be laughed at rather than that he'll be seen as a monster.
Eventually, having once more reviewed the circumstances, he reached the final conclusion that he should most nearly hit the truth in supposing Madame Podtochina (wife of the Staff-Officer, of course—the lady who wanted him to become her daughter's husband) to have been the prime agent in the affair. True, he had always liked dangling in the daughter's wake, but also he had always fought shy of really coming down to business… Yes, the truth must be that out of revenge the Staff-Officer's wife had resolved to ruin him, and hired a band of witches for the purpose, seeing that the nose could not conceivably have been cut off. (2.96-97)
It's interesting that Kovalev immediately gets into some super-paranoid conspiracy theories about what happened to his face. Some of it is probably some guilt about the daughter situation (you know, why isn't he putting a ring on it if he likes it?), but there's also a sense that the whole world that he lives in is kind of a mess of ulterior motives and hidden agendas. When you look at it that way, the idea that this is all part of some big plot against him doesn't seem so far-fetched.
Feeling, somehow, very nervous, he drew the mirror closer to him, lest he should fit the nose awry. His hands were trembling as gently, very carefully he lifted the nose in place. But, oh, horrors, it would not remain in place! He held it to his lips, warmed it with his breath, and again lifted it to the patch between his cheeks—only to find, as before, that it would not retain its position.
"Come, come, fool!" said he. "Stop where you are, I tell you."
But the nose, obstinately wooden, fell upon the table with a strange sound as of a cork, whilst the Major's face became convulsed.
"Surely it is not too large now?" he reflected in terror. Yet as often as he raised it towards its proper position the new attempt proved as vain as the last. (2.117-122)
The nose has been out in the world and it's gotten too fancy for Kovalev. Or something. Basically isn't this just a way over-heightened version of that weird feeling you get when you try to put on clothes that are too cool or too whatever—just not quite right for you somehow? It's like your body isn't really just your body but an extension of your personality. And what if they don't match? Then bye-bye cool new version of yourself.