Is a haircut on Rosh Chodesh Iyar permissible?
ב"ה Shalom The Shulchan Aruch (אורח חיים סי' תצג) says outright that those who have the custom to take a haircut on Rosh Chodesh are mistaken. The Mishna B'rura explains that there are two primary minhagim in in regard to the days one is not allowed to take a haircut during Sefirat Ha'omer due to the death of the students of R. Akiva. One custom is not to shave from the the second day of Pessach until Lag B'omer (or the 34th of the Omer according to the Sephardi custom) and from then on they may shave. The other custom is that one may shave from after pessach until the second day of Rosh Chodesh Iyyar but then would not be allowed to do so until the end of the omer. According to both customs the amount of days of not shaving are 33. So if one takes a haircut or shaves on Rosh Chodesh Iyyar and then shaves after Lag B'omer, he is taking advantage of the leniencies of two minhagim which contradict each other and therefore this is an incorrect practice. Furthermore, there is another custom mentioned in Sefer Chassidim (48) not to cut one's hair on Rosh Chodesh the entire year. It is brought down in the Mishna B'rura (סימן ר"ס ס"ק ז'). Not everyone follows this minhag, for the very fact that the above Shulchan Aruch only says one may not cut their hair on Rosh Chodesh Iyyar, which indicates that there is no problem on a different Rosh Chodesh. However, the Radbaz (ח"ב סי' תרפ"ז)[died in the Tzfat 1573]), mentions that for those who are extremely irritated by not shaving they may shave on rosh Chodesh Iyyar. The Yalkut Yosef (קיצור שולחן ערוך ילקוט יוסף סי' תצג) mentions, that those who follow the custom of the aforementioned Sefer Chassidim would shave erev Rosh Chodesh. However, the Yalkut Yosef adds that it is most worthy that a person uphold the custom of not shaving even then. On the other hand, If Rosh Chodesh Iyyar were to fall on Shabbat, the Mishna Brura (סי' תצ"ג ס"ק ה) says that since there is a combination of Shabbat and Rosh Chodesh 0ne would be allowed to shave on the Friday just before in honor of Shabbat. In short, answering your question, following the Shulchan Aruch and Mishna Brura, one is not allowed to shave on Rosh Chodesh Iyyar which is the more prevalent practice. All the best. Happy Yom Atzma'aut.