does the word for neighbor end in the masculine because you're referring to >your< neighbor and in this instance you're masculine?
There is nothing masculine in this sentence.
Judy is a woman so the "neighbour" is feminine: جارة.
The 1st person possessive suffix ("my") doesn't have a gender, it's always ي (-i) whether I am a man or a woman.
So we add the genderless suffix to the feminine noun.