While I can't answer for all of Asia, I can provide some insight into China, which was the dominant maritime power in Asia at the time of the time.

So, as for the first part: when did China become aware of the New World? It would have happened relatively quickly. The Portuguese had been steadily making progress towards Asia around the same time as Columbus' voyage. Jorge Álvares was the first Portuguese explorer to reach China in 1513, where they established a small colony called Tamão somewhere along the Pearl River Delta, which was evicted by the Ming Navy in 1521. Rafael Perestrello, the cousin of Christopher Columbus, led a trade mission to Ming China in 1516-1517 but was never allowed to meet with officials. Ming unfriendliness was partly because of reports of Portuguese soldiers acting wildly, and partly because they were aware of the difficulties that other Asian rulers were having with the Portuguese. For example, the deposed Sultan of Malacca, which had been conquered by Afonso de Albuquerque in 1511, had previously appealed to the Ming Court for assistance.

Nevertheless, Chinese traders did business with the Portuguese. Later missions, for example, Gaspar da Cruz in 1556, were able to meet Ming officials. Gasper also wrote an influential account of his travels, Tratado das cousas da China (Treatise on things Chinese), published in 1569. His travels were facilitated by Chinese who spoke Portuguese rathe than the other way around, suggesting that some Chinese had specialized in dealing with Portuguese traders by this point. By 1571, when the Spanish founded Manila, the Chinese could not help but be aware of the Spanish empire in the New World, as it became the source for a massive influx of silver into China. In The Search for Modern China, Jonathan Spence describes how China was part of a "global network" centered on "the fantastic silver riches of the mines in Mexico and Peru, which were being exploited under royal license by the Spanish conquerors of those territories." Chinese traders sold cloth, silk, and other valuable Chinese goods for silver from the New World. The flow of silver duly impressed the Wanli Emperor of Ming China (r. 1572-1620), who sought the advice of Matteo Ricci, a Jesuit priest, on matters of astronomy and geography. We can definitively say that the Chinese court was fully aware of the details of the New World in 1602 when Ricci produced the Kunyu Wanguo Quantu (坤輿萬國全圖), the first Chinese world map to show the Americas (whatever Gavin Menzies says). The map was exported to Korea and Japan as well, so it is probably safe to say that at least in elite circles, knowledge of the New World was reasonably widespread at that point.

The second question is more challenging – why did China not colonize the New World if they knew about it and knew how lucrative it was for the Europeans? This gets into a serious historiographical debate about why (and when, and how) Europe pulled ahead of China economically, technologically and militarily. Most Western development theorists (Marx, Weber, Polanyi, Rostow, Braudel, and Wallerstein, to name a few) traditionally argued that it resulted from some combination of Protestant ethics, cultural peculiarity, geography, and the nature of capitalist development that led European states to develop colonial empires and undergo an industrial revolution, two processes that were intricately linked. Europe's rise was depicted as effectively pre-determined, as was its colonial domination of the New World and Asia.

Starting in the late 1990s and early 200s, some historians began to move away from explanations that rely on culture or deterministic factors. In "ReORIENT: Global Economy in the Asian Age" (1998), Andre Gunder Frank argues that Europe's rise was largely a matter of chance, circumstance, and the specific pressures and incentives that arose from both China and Europe's individual political, social, and economic systems. Europe faced endemic competition and economic pressure that was absent in China, and a number of them sought the novel solution of seeking new lands for economic exploitation. Europe’s rise resulted from colonial powers inserting themselves into an Asian-dominated global trade network that already existed and was centered on China; a process Asians did not notice until it was too late. Asia’s agrarian land empires were also able to expand overland in ways European empires could not. They did so in the same period that the Europeans were conquering their colonial holdings. The so-called “Gunpowder Empires” (the Safavids, Ottomans, and Mughals) and both Ming and Qing China conquered considerable amounts of new territory during this period, and so they did not need to seek new methods of expansion. They also refined their own systems of agricultural economic exploitation.

So what does that mean for the question of “why did China not expand overseas”? Well, first of all, as mentioned, they did not need to. They had plenty of opportunities to expand overland. The consensus of historians is that they chose the most profitable course of action for the political and economic system they had developed. Europe did so as well, which led to an acceleration that allowed them to overtake Asia. Furthermore, they already had access to the fruits of the New World in the form of Spanish silver via Manila. They were content to sell their products, which were in high demand, and reap the rewards. They were also unable to see the long term consequences of the disruption to the traditional economic system caused by the major influx of silver, a factor in the collapse of both the Ming and Spanish empires.

The last of the Ming exploratory treasure voyages ended in 1433. Historians disagree as to why, especially as they did not seem particularly expensive and in some cases did provide a small profit. The most convincing explanation to me is that it was the result of court politics and the influence of rich and powerful elites who for personal and political reasons were opposed to further voyages. Regardless, such trips didn't provide the same return on investment as they did for Europe. It is hard to say whether it would have been to their benefit to continue to do so, but even after learning of the New World, Chinese elites likely felt that it was easier to benefit from Europe’s colonial empire than to establish far-flung colonies overseas.

It’s also important to remember that Chinese commercial activity overseas was traditionally carried out without state assistance. Major communities of overseas Chinese could be found throughout Southeast Asia and the Pacific islands, where they made their living as merchants and laborers. Following similar patterns, Chinese people did eventually make their way to the New World, attracted for much of the same reason as many European immigrants. Chinese merchants, fishermen, sailors, and laborers could found in the Spanish Empire, especially between Manila and Mexico. By the mid-18th century, there were a handful of documented Chinese communities in California. As Chinese presence in America has mostly been studied from the perspective of Chinese-Americans, less is known about whether Chinese communities existed prior to the mid-1700s. Still, it is well documented that there were communities after that, and steady waves of Chinese immigration began to the United States in the early 1800s.

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Edit: Minor change to the second to last paragraph on Ming treasure voyages to improve accuracy.