This is actually easier than it might have been.
To put something that happened in the past in context, for instance, requires a fair amount of knowledge about the period of the event, the characters involved, their backgrounds, families, the pressures on their lives, and the analytical ability to extract the important pieces and the relationships they created, and then the writing ability to frame the event, and all that body of knowledge that very few people around you would actually ever pay attention to, in a way that grips them long enough and well enough to read through your work.
But placing a quote, and especially a word, from a literary work into context only requires you work with the book itself. It can be tough, especially if you want to go past the obvious. But seldom too complicated and seldom requiring weeks of research!
For instance, placing "Rudolph with your nose so bright, won't you guide my sleigh tonight?" into context could be as simple as simply refreshing the readers' memories of the fogginess of that Christmas Eve and the idea that Santa probably needed to be able to see where he was going to deliver presents to the world, and then pointing out that Rudolph's glowing nose might have been an entirely adequate solution and due to there being no other solutions with time running out, everyone was likely filled with joy to hear Santa realize the solution and that the world can be filled with joy one more Christmas. Nothing needed past understanding the flow of the story and the simple needs and motivations of the characters involved.
Of course, one could go past those simple things and think on Rudolph's ostracization, the way his childhood was crushed by it, his exile, how he found and made friends, managing to overlook all their flaws to see their good features. One could consider the clearly basic "user" nature of Santa and utter "what value do you have to me" nature of all the other reindeer, the reindeer hatefulness, Santa's tacit acceptance of it. Instead of seeing Santa and the other reindeer as, in one huge and completely unstated leap, coming to see the error of their ways and to utterly value Rudolph now and forever, one could be far more realistic and see their venality and "user" nature. These things could lead one to place the quote in a more realistic context, one that isn't so joyful and uplifting and leads to a future past the story's end that isn't likely to be a pretty one.
Either is an example of placing a quote from a literary work into context. I don't mean to give the impression one must look for a context like the second one. The obvious one is usually the intended one. Of course, the finer the piece of work, the more likely there is plenty beneath the obvious. Teachers sometimes are more impressed by students who can suss such things out. Or not. Regardless, you yourself will learn more, about the work, and about life itself, looking past the obvious. That will only help you in later schooling and in life.
The assignment lets you choose the quotes, and in the next line, the words, that you will use. For the assignment I imagine you should stick to choices that you feel you fully understand. Then just give their context, like above!
Oh, don't pick "No Country For Old Men" as many people I know who are experienced enough with life to expect to understand it flat out just don't really get it. As a student, you're likely young, and therefore inexperienced at life so would have to work much harder to understand the book enough to do the entire assignment. Of course, playing the best and not midnight basketball junkies is a large part of how Michael Jordan got so good. It might be rewarding to choose that choice. But summer is running out and and time could be important here.