I have a question on English preposition "by".?

No, you can't use By in this case. He was elected IN 1960, it's an action that was completed in the past. If you said he did something "by" a certain time, it's like you're telling a story about how he had achieved certain things after a number of years. By the age of 35, the singer had earned 10 gold records. By October 1st, all the leaves had turned brown.

This may be a hard point to understand, but if you said "By 11 o'clock, all the kids were asleep", that means some fell asleep earlier, but at 11 o'clock, all of them had fallen asleep.

If instead you said "The kids fell asleep at 11 o'clock, that means they all fell asleep at exactly the same time, 11 o'clock.

So if someone was elected, which is a decisive thing that happens on one day and is finished, you say when it happened. In 1960, he was elected. At 4 o'clock he was arrested, etc.

The only exception is if this sentence is part of an ongoing biography of Kennedy that also explains all his achievements. You could say "He was born in 1920, he grew up in Boston, and by 1960 he had been elected the first Roman Catholic President. In this case you are describing what he did "by" a certain age or point in his career.

I hope this helps, it's a difficult question to explain properly

You can say "by" but the verb tense is wrong.  It would have to be phrased like this:  "By 1960, John F. Kennedy HAD BECOME the first Roman Catholic president."

If you say "became", it would be:  "In 1960, John F. Kennedy became the first Roman Catholic president."

BY in this case could only be use if the previous statements include something/s done in the previous years. 

No. If you know the precise date when something happens you would say it happened IN that year. (Presidential elections are a matter of record).  What you can say is something like: BY the time JFK was elected in 1960, it had become acceptable for a Roman Catholic to be President, since the matter of acceptance is the thing that is less precise.