جَاءَنِي الْعَصَا وَغُلَامِي: The stick and my servant came to me.
وَ: coordinating conjunction.
غُلَامِي: coordinated with الْعَصَا; hence nominative with estimated ḍamma on mīm, hidden because that position bears the kasra required before possessive yāʾ.
غُلَامِ: noun in an iḍāfa; ي: possessive pronoun, in the genitive position.
Remember: The kasra on mīm does not turn the subject into a genitive noun.