Does eating before Alos hashachar on Rosh Chodesh help for not fasting on Rosh Chodesh?
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.