A teacher could afford Brothers Karamazov, but there are a number of challenges to overcome. Much of Russian literary fiction of the 19th century was published in journals rather than books, and distributed to subscribers (theoretically bounded only by the limitations of the post office, but in practice concentrated in urban areas). In the 1880s an industrial laborer could expect to earn 70 kopecks to 1 ruble a day, and books ranged from 1 ruble to 10 rubles depending on circulation and printing quality-- a teacher can afford a book, but it would be a splurge. Depending on the oversight of the Zemstvo the teacher would be unable to provide instruction from the novel, and most of their students would come to them illiterate; local schools were sponsored by the Zemstvo, in turn receiving church support. Texts that challenged or complicated Church doctrine (consider Alyosha's response to the Grand Inquisitor) simply wouldn't be taught, at least in a public setting. Universities would be a more natural place for novels, though in the 1880s there were only nine institutions in the Empire.

Brothers Karamazov was serialized in the Russian Messenger from 1879-1880, shortly before the author's death. Subscribers numbered in the low-to-mid thousands, primarily in Petersburg and Moscow, but accessible in secondary and tertiary towns; you may or may not be able to subscribe from a village, depending on your regional infrastructure. If you are a rural teacher you will need to travel to a city to purchase books and backordered journals, but in general you possess the ability to do so.

Bookstores were more uncommon. Kharkov, with a population of 50,000, had only a handful of bookstores in this period, primarily focused on student textbooks; Irkutsk and Tomsk had just one each. Petersburg and Moscow certainly had booksellers--Petersburg as early as the 1730s--but long texts underwent censorship retroactively, so a book might be published but then banned (indeed Dostoevsky's works were banned for instruction in the 1880s, and particular passages of BK censored for recitation)

Public libraries ostensibly appear in the liberal 1830s, but without state funding and supplies they were functionally inaccessible; by the 1860s, paid libraries (on a subscription basis), were relatively common, but often only had a few hundred books. Literary fiction again played a very small part, which brings us to the question of literacy: only around 20% of Russians were literate in 1880, and most who were had been taught using the Bible and hagiographies. Fiction didn't really factor into secondary education, and was perceived circumspectly by both the rural populations and regional administrators.

In Dostoevsky's first novel, Poor Folk, we see a rather fanciful view of how lower-middle-class urban workers engage with books, "squandering" money on them; they are only occasionally accessible in the market (Gostiny dvor), but when they are available, they are discounted; and the father of the tutor Pokrovski purchases a set of eleven books that he had "long been saving for," relying heavily on his son's income. (Other books are noted to be one, two, or three rubles, and the 'nicer' books he cannot afford at all).

In general, books were a luxury, and high fiction was generally restricted to the urban, affluent classes. By the 1880s there was a broader readership, and a schoolteacher most likely had the means to access and purchase something like Brothers Karamazov.

Now that another redditor answered about the Brothers Karamazov, I found some information on teacher's salaries.

First of all, we need to understand that there were many types of teachers, and their earnings varied depending on experience, gender, where they taught and where they where located. There were many types of schools in the 1880s, but the main were: "народные училища" or "people's schools" (schools for the poorer groups like the peasants and craftsmen, which were 90%+ of the whole population), which were located mostly in the countryside and offered just a tear or two of primary education, the "городские училища" or "county schools", for the more wealthy people like traders, wealthy peasants and townsfolk, which also offered just a few years of education, however accepted children who could read and knew basic arithmetics, so were a bit more advanced, and, finally, the gymnasiums. These were the elite private schools for the nobles, which, of course, offered the best education. It should also be mentioned that all noble children where homeschooled until around 8-14 years old.

Now, to the teacher's salaries. Let's start with the gymnasium teachers. The median salary was around 1200-1500 rubles per year. These teachers, however, were devided into tutors of the arts and the sciences, and the former were valued more and could earn twice as much as the latter. Interestingly, teachers of male gymnasiums had a fixed income based on their years of experience, and teachers of female gymnasiums had salaries that were counted based on the amount of lessons they had throughout the year. Also, teachers of gymnasiums could expect different bonuses. For example, language tutors were given an extra 100 rubles for checking student's notebooks, also, teachers with families could earn more. These standards were set by a law in 1871, and salaries didn't increase for around 30 years, so you could say that they decreased considering inflation.

It is much harder to count the salaries of lower-rank teachers, as they varied highly by region. By 1896, a third of the teachers earned less than 200 rubles per year, and a fourth- under 100, which is Incredibly low (but these were mostly teachers who worked part-time and for only about half a year, so they had to get many other jobs). This situation was addressed by the government in the 1900s, but it didn't help too much and a lot couldn't be done because of the revolutions in 1905 and 1917, as well as the Russo-Japanese war in 1904-1905 and WW1 in 1914-1917. However, teachers received pensions and donations from the Government and even more from private funds, as philanthropy was very popular amongst the richest nobles and traders, although not everyone could get them.

So, some elite teachers could afford books, but most couldn't.