With the phrase "good night" do you have to use an apostrophe before the word "night" if you are to omit the use of "good?"
You don't need an apostrophe if you just say "Night." If you clip part of the "good" and say "g'night" then you need an apostrophe. This is actually how the phrase is heard most often.
I sense your discomfort at just using the single word as a sentence.
He rolled over and turned off the light. "Night," he whispered, and kissed his wife on the cheek.
That's a perfectly understandable and even grammatical expression. The "good" is understood, because it is such a common expression.