Obviously this kind of question is dependent on where your "random person" is (an uncontacted tribe in the Amazon is quite different than your average American), but if you are talking about the Western world, yes, atoms would have been part of both standard educational curriculum, the general zeitgeist, and popular knowledge by 1960, for sure. The Google Ngram viewer suggests that the word atom was more frequently used in print in 1960 than it even is today, and the adjective atomic even more so — no doubt in connection to the very real interests at the time that people had in nuclear weapons and nuclear power. To give a sense of how common the word was, compare it to another common word.

Now, a deeper question would be: what would they understand that word to mean? What would the depth of their knowledge be? Would they be able to draw a schematic atom with subatomic parts? This is a lot harder to answer. But again, you have to keep in mind that this was a time in which the atom was a core part of how people saw their political world — because of nuclear technology — and so was probably more commonly discussed than a lot of other scientific concepts. (It is interesting to compare the frequency in print of "atom" against "DNA" — DNA was not all that commonly used a term in 1960, but would become an immensely commonly used one by 1980).