Shabbos Rosh Chodesh Laws and Customs
Sources: Rabbi Yaakov Goldstein Halacha Shiur (Daily Halacha) Shulchan Aruch Harav - Rav Shneur Zalman of Liadi (https://shulchanaruchharav.com/shabbos-rosh-chodesh-laws-and-customs/#_ftn4)
When Rosh Chodesh falls on Shabbos, one is to prepare an extra dish for the Shabbos day meal in honor of Rosh Chodesh. (Elya Raba 419:1; Chayeh Adam 117:2; M”B 419:2)
Some Poskim rule that one is to celebrate the Rosh Chodesh meal on Sunday. This is because, in any event, one has a festive dinner on Shabbos; hence, there is no recognition for the meal of Rosh Chodesh. (Rif, Rosh)
Some Poskim suggest that one is to continue the third meal into the night of Motzei Shabbos in honor of Rosh Chodesh, and in this way, one also fulfills the Mitzvah of Rosh Chodesh. (M”A 419:1)
Some Poskim rule that when Rosh Chodesh falls on Shabbos and Sunday, one is to follow the advice of the M”A (ibid) and continue the third meal of Shabbos into Motzei Shabbos, as well as have a festive meal the next day on Sunday. [Nimukeiy Orach Chaim 419:1]
Other Poskim wrote to increase the Melaveh Malka meal in honor of Rosh Chodesh. (Siddur Yaavetz brought in Ben Ish Chaiy ibid and Kaf Hachaim 419:3)
When Rosh Chodesh falls on Shabbos, one says Ritzei and Ya’aleh Veyavo.
On Shabbos Rosh Chodesh, during Mussaf, one adds to the fourth blessing of Shemoneh Esrei, the paragraph of Ata Yatzarta, which discusses both the sacrifices of Shabbos and Rosh Chodesh. One concludes the blessing with “Mikadeish Hashabbos Yisrael Veroshei Chodashim.”
Av Harachamim and Tzidkasecha are not recited on Shabbos Rosh Chodesh.