The correct answer is A.
The phrase that we want to focus on is the one either in between commas or dashes, which is called an aside. This means that if we were to take this phrase out of the sentence, the rest of it would still make sense.
For example, if we take out "between 1920 and 1957" from the first option, "Eugene O'Neil won the Pulitzer Prize four times for his plays" still makes sense. On the other hand, if we look at the second option and take out "four times between," then "Eugene O'Neil won the Pulitzer Prize 1920 and 1957 for his plays" does not make sense.