Hi guys,
Not for nothing is the city a sister city of New York can be rephrased as
The city is not a sister city of New York for nothing. The meaning is that there is, indeed, a reason.
The double negative adds emphasis. For example,if someone asks me to help them with their English, I might say I don't work for nothing! (Not, alas, a principle that I follow on this forum)
The inverted word order adds a rather rhetorical flourish and style to the sentence.
You'll find a few examples of this via google, eg
Not for nothing do many established professional writers consider that it is harder to write a good short story than a novel.
Best wishes, Clive