Chinese culture had a very large presence in the imaginations of 17th and 18th century Western intellectuals, but as /u/Forma313 said, Western portrayals of China were rarely accurate. It was during this period that what historians call "Sinomania" came into fashion throughout the West. This was fueled by Jesuit reports that presented China as prosperous and enlightened. The Chinese government was described as a well-ordered, meritocratic bureaucracy. Matteo Ricci wrote that "the entire kingdom is administered by the Order of the Learned, commonly known as Philosophers."

The enthusiasm for China can partially be attributed to how European philosophers used China as a foil for the problems of the West. It was very common for Western philosophers to attach Enlightenment values to their idealized China. For example, they believed that China had a system of checks and balances that would allow government mandarins to have (according to Olfert Dapper) "undaunted freedom and confidence in the admonishing of their kings and emperors, when they saw them wander from the way of virtue." Voltaire, who was probably the most vocal of the sinophiles, would use China as a tool for attacking the Catholic Church. He theorized that the Chinese were deists who based their religion on reason instead of faith. The existence of Voltaire's romanticized Confucianism was cited as proof that a civilization could thrive without the moral guidance of the Church.

This was just a quick summary of the sinomania fad, and I skipped over other prominent sinophiles such as Leibniz. For a more in-depth history of sinomania, I would recommend reading Oriental Enlightenment: The Encounter Between Asian and Western Thought by J.J. Clarke and China and Europe: Images and Influences in Sixteenth to Eighteenth Centuries, a collection of articles edited by Thomas Lee. There are also several excellent articles on the sinomania hosted on JSTOR.

tl;dr with access to the right books his knowledge could have been quite extensive, though not necessarily accurate.

Portuguese traders had been present in Macau since the early 16th century. Around the same time Jesuits (of various European nationalities) took up residence at the court in Beijing, and stayed there, at first serving the Ming emperors, and later their Qing successors. The main thing they impressed their hosts with was their knowledge of astronomy, beating the local competition in making predictions. (though IIRC by the 18th century they mainly relied on imported almanacs). Their writings on China, including a book on the Manchu conquest, were published in Europe. At least in part thanks to their publications and correspondence with the French court Chinoiserie became popular in Europe in the 18th century.

Between 1624 and 1664 the Dutch East India company had a presence on Taiwan, which was then not yet part of the empire, and traded with merchants from the mainland.

From the end of the 17th century onwards other European traders began regular voyages to Canton. To be honest i am not sure when American traders started to make the trip, or how widely available European books were in the 13 colonies. But, here is an example of a 17th century book on China. Orignally published in Dutch and translated into English, though it contains segments from other European publications.

As for Japan, there had been European merchants there since the 16th century (the Portuguese got there first of course), but from 1624 only the Dutch remained. I'm fairly certain books were written about Japan as well, but none spring to mind, alas.

What does Ichabod say about Japan and China BtW?

To add to what others have mentioned, this would have been right around the time that American merchant contact with China really began. Trade started in 1783 as the economy officially became free from British control and many ships used in the war, like the Empress of China, were available to be converted into merchant vessels (source). Northeastern port cities would have seen a growing number of businessmen invested in the trade, many of whom would have actually traveled to China.

If Ichabod Crane was a school teacher from Connecticut living in New York, it's not impossible that he would have come into contact with a merchant or someone with first-hand knowledge of China. That said, the likelihood of him having any sort of in-depth understanding of the country or its culture is low, as others have already outlined. Even the merchants themselves, operating under the Canton System, were not allowed outside of the city of Canton onto the Chinese mainland (source).

I'd be interested to know what kind of information he claimed to know in the show.

If you get a chance, check out the Peabody Essex Museum in Salem, MA, to get an idea of what some Americans at that time might have seen and understood about China. The institution was originally founded as the East India Marine Society in 1799 and it's done a good job of preserving artifacts from the early trade with China.