To my understanding Europe was unaware that there was a continent across the Atlantic. Given the relatively narrow Bering Straits and the Aleutians, was the same true for Eastern Asia and the Pacific?
Not to my knowledge. Most of East Asia's (specifically China) limited exploration was centered on traveling West (with well-documented expeditions through Central Asia and even to the Middle East), which is intuitive if you think about it as, of course, the Pacific Ocean is significantly larger than the Atlantic.
If you dig around this topic, you're almost guaranteed to find news articles about how the Chinese explorer Zheng He supposedly landed on and created an absurdly detailed map of the entirety of the Americas. Of course, this is pretty much guaranteed to be a hoax.