"I" is used when the speaker is the subject of the sentence:
"I kissed Eve."
"Me" is used when the speaker is the object of the sentence.
"Eve kissed me."
Strict grammatical logic says that the same is true when answering questions.
"Who kissed Eve?"
"I did." or just "I."
However in common usage, going back centuries, people frequently use "Me" as a single word answer where "I" would seem logical. It's so common that it can't be considered wrong:
"Me."
..
"Who did Eve kiss?"
"Me."
By way of illustration; the classic nursery rhyme:
"Who killed Cock Robin?"
"I," said the Sparrow, with my bow and arrow"
Adding "not" changes nothing.
"Who kissed Eve?"
"Not I" - agrees with the formal, and still acceptable grammar
"Not me" - agrees with common usage
Both are fine. "Not I" is perhaps a bit ostentatiously literate
"Who did Eve kiss?"
"Not me." -- never "Not I"
Just as with that old chestnut less versus fewer, it's a one way street. People rarely say fewer when less is "correct" (except sometimes deliberately, to annoy). People rarely say "Not I" when "Not me" is "correct".
Bottom line: if you always say "Not me", you'll never be wrong. But you miss out on the chance to be extra rigorous in your grammar.