When reading this poem, I hope you can see When we should use "Not I" or "Not me." The context determines what should be said. I hope I don't have you scratching your head.
If somebody asks, "Who's a good dancer?" Please do not let "Not me" be your answer. If you are NOT, then you should reply, "I'm not," or simply respond with "Not I."
However, if somebody asks, "Last week, Who(m) did Marisol kiss on the cheek?" IF "She didn't kiss me" is how You'd answer, "Not me" is sufficient right now.
"Who wants this book?" "Not I" you'd say If you do not want the book today. "Who(m) did Mario call a ****?" He DIDN'T dis YOU? "Not me" then would work.
Didn't you learn "Not I" as I did When being read to as a young kid? "Who'll help?" asks the little red hen with a cluck. "Not I," say the dog, the cat, and the duck.
As you can see, it all depends On what one means--on what one intends. Whichever expression you use will suffice, But one's always clearer and much more precise.