Clearly the way in which the narrator is shown to be a rather limited individual who is very unsympathetic and lacking in empathy is revealed at key points in his narrative. Consider, for example, the numerous faux-pas that the narrator makes in his conversation with Robert, much to his wife's chagrin and disgust. One example of this is when he struggles to find something to say and asks Robert about his journey:
Then I wanted to say something else, small-talk, about the scenic ride along the Hudson. How going to New York, you should sit on the right-hand side of the train, and coming from New York, the left-hand side.
We see in this quote the narrator's own inarticulate nature and his desire to make "small-talk," and how this desire leads him to draw reference to Robert's blindness. Such an example, combined with others, such as when the narrator turns the television on when he gets bored of the conversation, shows him to be inarticulate and isolated and also very unsympathetic and lacking in empathy. It also can make us feel slightly sorry for him, as he desires to be able to communicate, but is lacking in the skills of building and establishing relationships.
In answering your question, the first thing we should do is clearly define the terms and then, emerging from ambiguity, an answer will begin to present itself.
Style is the manner in which the author tells the story. As for character, here is where we hit a snag, because the word sets itself up as a double-entendre. By character do you mean the personality of the author? Or do you mean the key person the author is attempting to depict in the plot?
Actually that question is revolved by the word narrator, which makes it clear that the "author" we refer is an entity that the true author of the book uses as the medium through which the story will emerge.
Think of having a conversation with someone and everything will make more sense. You can tell what kind of person someone is after talking to them for a few hours. If you're really intuitive, you can tell after a few minutes.
The same is true for a skilled reader who is familiar with the literary field. Sometimes the character of a narrator will be clear in the first few paragraphs. Other times it may take us awhile to really develop an understanding of what they stand for.
Think of the expressions they use, the quality of their language. Is it coarse, or refined? Poetic? Rambling? What emotion-related words do they use to describe events in the plot? Perhaps they are totally distanced and unbiased.
Sometimes we learn more about a narrator's character from the observations they make about others than by what they say about themselves. Consider as an example, Nick Carraway, the narrator in Fitzgerald's The Great Gatsby. We never really learn anything about Nick, but based on the scenes he selects to reveal to the reader, we can see what catches his attention by being contrary to the "norms" of his own thoughts and emotions (namely the erratic, reckless behavior of people who suddenly have come into wealth).
I would suggest paying attention to a few key things. What does the narrator chose to describe? How does he react to these events? What type of expressions and/or colloquialisms does the narrator use? What type of references does he make (this last one will take some cultural awareness)? Once you start picking up on these things, other categories will begin to present themselves for inquiry.
Overall, treat the book as a conversation with the narrator and it will come a lot more naturally.