While I'm aware that "people of Asia" is a huge scope, I'd be interested in pretty much any account of any people in the continent of Asia's reception to the 'discovery' of the Americas.
Sorry for the copy & paste with some complements from my previous post in: How did the Russians not discover the New World before anyone else when they're so close together?
+++
While much more can always be said, the following posts of mine might be interesting to you:
+++
As for the "Chinese (including Yuan)" expansion in the Far Eastern coast (and the Amur River area) and its geographical limit, the following previous posts in this subreddit will offers you some basic information:
+++
In short, in addition to Japan, Yuan and (early) Ming China also certainly regulated the trade/ tribute collection from the Ainu (and other hunter-gathering peoples in NE Eurasia like the Nivkh at their north-eastern outpost, but even the geographical expansion of Ainu's culture and trade sphere into the Northern Kuril and the southern end of Kamchatka Peninsula came relatively late, around the end of the 15th century.
As for basic outlines of Ainu's involvement with Japanese authority in the south in the 15th and 16th centuries, please also refer to Was there any significant Ainu involvement during the Sengoku Jidai?, answered by /u/ParallelPain.
+++
I know these previous posts are bit weak in response to OP's original question, especially "How did they initially react" part, but I'd like to argue that the "Discovery of the New World (both American Continents)" itself was not so paradigm-breaking for the the 16th and 17th century Asians than the Europeans who had been familiar with the concepts of three continents of the Old World (Africa, Asia and Europe) for long.
Concerning the existence and the formation of the Asian (Chino) communities in Mexico City and their components after the establishment of the Manila Galleon Route in the Pacific (1571), please also refer to /u/drylaw's and my post in: How interconnected was the world of 16th - 17th century Mexico? Was it possible to encounter destitute Japanese samurai and West African maroons as depicted in 1493 by Charles C. Mann?.
Local archive from New Spain (Nueva Espana) also sometimes reveals some sporadic entries of East Asians - mainly taken probably involuntarily as slaves, but a small number of either freed ex-slaves or free settlers. To give an example, the following breakdown of Asian-origin slaves in New Spain from 1565-1673 by Oropeza Keresey includes 4 Japaneses and 2 Chinese of 93 Asian slaves in total (contra 50 from India and 31 from Philippine) (Oropeza Keresey 2011: 27, Cuadro 1).
Additional Reference: