For example would a Japanese aristocrat know about the major Chinese dynasties, the major emperors, poems, political system, etc. Also how familiar would a Japanese peasant be with Confucius, Mengzi, and the other major Chinese philosophers and their works? How much access did medieval Japanese society have in general when it came to factual knowledge of China? For example did the Japanese have a rough idea of what the geography of China was like, what kind of languages they spoke, etc. Did the Japanese have maps of China?
In the Nara and Heian periods (roughly 750-1190) China was the THING. If you didn't speak the language or write in it, etc, then you weren't exactly worthy of the title 'noble'.
This is only the nobles who live in the Imperial Capital, however. The warrior families living out to the east and south would've had less knowledge of China, although most Shugo, Daimyo, and Samurai throughout the entirety of Japanese history would've been familiar with Confucius, Mencius, Lao Tzi, and Buddha--or at the very least just with their names if nothing else.
The Kamakura Shogunate, established roughly in the late 1180s and early 90s, shifted the de facto capital further east and removed many of the Chinese Cultural aspects. The next shogunate--Muromachi, or Ashikaga--moved back to Kyoto and the impact of China rose again among the nobility, and this time among the warriors as well. The peasant overall would not know much, if anything, other than perhaps the name.
Those most familiar with the geography on the other hand would most likely have been monks--although I'm not entry certain so feel free to research on your own. Monks and other pilgrims and scholars had routinely been sent to China in a cycle--a few years of sending, then the emperor has a change of heart and bans contact, he dies and then the next one reopens, etc. Therefore they would've been most familiar as they had actually been to the place. They could've also studied from maps.
If you'd like more info check out the histories of Japan by Samson, Morris, Turnbull, Friday, and Resichauer. For the Chinese aspect look into Fairbank and Spence. You could also check out some primary sources.