There's a concept of early Shabbos. A person can accept Shabbos early, pray Shabbos Maariv before dark (according to Rabbi Yehudah, who says you can pray after Plag HaMincha; I admit this one is less unanimous), say Kiddush, are prohibited in melachos, etc. The same is true for Yom Tov (with a few exceptions).
Is there such a concept as early Rosh Chodesh? I'm not sure how'd they'd accept it upon themselves. Perhaps by praying Maariv after Plag HaMincha? If someone prays Maariv after Plag HaMincha on Erev Rosh Chodesh, do they say Yaaleh VeYavo? If they then eat a bread meal, do they mention it in Bentching? Those who are accustomed to refrain from certain activities on Rosh Chodesh, would they then apply?