Super abridged version:

In the 1800s Britain fought two wars with China, the First and Second Opium War, and these largely resulted in the signing of what are referred to as the Unequal Treaties starting in the 1840s. The treaties significantly favoured foreign powers and hurt China, but with a weakening Qing there wasn't a terrible amount that China could do at the time. We can consider this the period of bullying. Missionaries became a common sight; Western powers had rights of extraterritoriality and people were getting justifiably cranky. This in part lead to the Boxer Rebellion, which was an expression of anti-foreign sentiment in China. This in turn lead to the Eight-Nation Alliance invading China to exert their influence a little more strongly. The Qing really didn't stand a chance at this point. Not long after, the dynasty fell and in 1911 the Republic of China was founded (and continues on today as Taiwan). Between that time and the end of the 1940s, the Nationalists (ROC) and the Communists were fighting with each other, with the Japanese, with warlords and so on.

So back to your question. Did the fall of Japan cause China's rise? This may be a matter of opinion. I would argue that no, it did not. Japan was not likely to have been able to sustain empire in Greater Asia as was the plan at the time. There was a lot going on back home in Japan, and the Communists were doing a pretty decent job at kicking ass and taking control. Thanks in no small part to issues within the Nationalist party, who were themselves somewhat thuggish.

Communist China did benefit from an alliance with the USSR, but the most productive period was short lived and China ended up feeling abandoned in the end.

Fast forward to the late 60s. Relations with the USSR are weakening as Mao is getting a little more eccentric. Coincidentally, it was in part the weakening of ties between China and Russia in the early 1970s that helped ease tensions between China and the US, helped along by Nixon's visit in 1972. Taiwan meanwhile was dealing with the White Terror, a period of martial law which didn't end until the late 80s, and really didn't end until 1995 since it was basically forbidden to discuss the period until that year's presidential apology. Meanwhile over in China, the cultural revolution is going on. The Red Guards were causing havoc, killing their teachers and ratting on their parents for being counterrevolutionaries. It really wasn't until the period of Reform and Opening starting at the end of 1978, also spearheaded by Deng Xiaoping, that the country really started to see the meteoric rise that's lead to today. Deng was a pretty major reformist and though his efforts capitalism was promoted. "To get rich is glorious" and "Black cat or white cat, it doesn't matter so long as it catches a mouse", he said.

I would argue that it was the Reform and Opening period that really prompted China to develop into an economic superpower, thanks also in part to a pretty damn effective means of state control and enough common sense to reform in whatever small steps are going to help its continued growth. There are all sorts of things that can be added to this regarding their management of their economy, but I'm not qualified to address it. So even though I feel like I understand it pretty well, I won't touch that part.

edit: fiksed sum typoes. also added hyperlinks in case you're curious about any of these points.