Does eating before Alos hashachar on Rosh Chodesh help for not fasting on Rosh Chodesh?
Answer:
No, this does not help.
For days on which it is forbidden to fast, which include Rosh Chodesh, one must eat specifically after Alos Hashachar; the time before Alos Hashachar is still considered the previous night, and therefore eating at this time will not be “breaking the fast.”
See Beis Yosef, Orach Chaim 418-9, and commentaries to Shulchan Aruch 418. It is clear from Se’if 3 that the obligation to eat (and prohibition to fast) applies after Alos Hashachar, which is when a private fast begins.
Best wishes.
