Question: Does the issue of cutting hair on Rosh Chodesh apply to a moustache as well?
Answer: Although from a Halachic perspective there is no reason not to cut hair on Rosh Chodesh, the view of Rebbi Yehuda Hechosid (Siman 48) was that one should not cut hair or nails on Rosh Chodesh.
Mishna Berura (Siman 260 s.k. 7) writes “there are some places that have the custom not to cut hair or nails on Rosh Chodesh even when Rosh Chodesh coincides with Erev Shabbos, as per the instruction of Rebbi Yehuda Hechosid”.
[It is interesting to note that in the context of cutting hair during sefiras ha’omer, the Mishna Berura (Siman 493 s.k. 5) rules that when Rosh Chodesh Iyyar coincides with Shabbos, it is permissible to cut hair on Erev Shabbos (because of the combination of Shabbos and Rosh Chodesh). Since Rosh Chodesh Iyyar is always two days, whenever Rosh Chodesh Iyyar coincides with Shabbos, inevitably Friday will be the first day of Rosh Chodesh (because Rosh Chodesh Iyyar does not occur on Shabbos and Sunday), and here the Mishna Berura does not mention the issue of cutting hair on Rosh Chodesh.]
Returning to your question, it appears that according to Rebbi Yehuda Hechosid, the issue of cutting hair on Rosh Chodesh applies to the moustache as well.