Literacy was widespread even before the rise of public schools. Beth Barton Schweiger, a University of Arkansas historian, estimates that in the South during the early 19th Century, over 80% of the white population was literate. While this level was lower than that of New England, it compared favorably with the literacy rates in Western Europe. She further notes that with the nonexistence of public schools and the limited number and cost of private schools, Southerners, even slaves forbidden from knowing how to read, were either self-taught or learned from their parents. In other words, they were home schooled. Diaries and letters reviewed by Schweiger emphasized reading as a means of bettering oneself, revealing a commitment to self-improvement that seemed to permeate the South.
It is likely that the situation in New York was similar, or even better, given the large numbers of New Englanders who had emigrated to the Empire State.
As to the level of literacy in the early 19th Century South, Schweiger has encountered a great deal of religious material -- from books of sermons to ecclesiastic newspapers, religious pamphlets and essays on church doctrine. The Bible itself is not an easy read: The New International Version, currently the most popular translation among conservative Protestants, is written at the 9th grade level of literacy. Most likely, the King James Version was at a similar level for the 18th and early 19th Centuries. Look at works like The Westminster Shorter Catechism, Jonathan Edwards' "Sinners at the Hand of an Angry God", or Pilgrim's Progress, staples among Christians of the same time frame. Having read those works, I am of the opinion that people who could understand them could master the writings of Hamilton, et. al.
Keep in mind, too, that, while New York was ethnically and religously diverse even in the late 18th Century, there was a common Calvinist theology in three of the four leading denominations in the state (Dutch Reformed, Presbyterian, and Congregationalist). As stated before, Calvinist theology emphasized the ability of and the necessity for the individual to understand and interpret God's Word. Thus, reading comprehension and ability would have been at least as high, if not higher, than in the South, where religion had a more emotional, revivalistic element.
Obviously, as urbanization and specialization increased, reading took on an increasingly economic and business related role. However, Americans of the country's first 250 years of existence were predominantly literate and had strong reading skills, due in great extent by a desire to know more about the Christian faith and to achieve self-improvement. No society is perfect and antebellum America had its share of faults, multigenerational human slavery being the most glaring. However, too many contemporary Americans fall for the error that the people who lived in our agrarian past were stupid, mimicing Karl Marx's slur about "rural idiocy."