The second mishnah in Rosh HaShanah says that "all the people of the world" are judged on Rosh HaShanah. How should non-Jews observe the Day of Judgment?
The nearest answer I can find refers to the prayers that Noahides can say on or before Rosh Hashonoh from “asknoah.org”
The suggested prayers contain a section from “Unesaneh Tokef”, “V’Khol Ma’aminin”, Psalm 24, and some prayers from from the booklet "Prayers, Blessings, Principles of Faith, and Divine Service for Noahides (5th Edition, pub. 20’13 by Ask Noah International)"
I saw an article where Rabbi Israel Chait addresses your question for Bnei Noach. I thought you might like his comments.
Here is a copy:
Noahides and Rosh Hashanna
Rabbi Israel Chait
What is appropriate behavior for Noahides on Rosh Hashana and Yom Kippur? I want to be very careful not to transgress by doing more than is permitted, not creating festivals for myself. But it seems to me – please let me know if I am mistaken – that at least Rosh Hashana is relevant to the whole world and perhaps I should mark it in some way. And finally, I would like to know if there are particular prayers from the Siddur that are permissible for the Noahide to pray.
Rabbi Israel Chait:
The Noachide should know that he too is judged on Rosh Hashanna by God just as the Jew and the rest of mankind. He therefore should pray all the prayers that the Jew prays, as he too is loved by God and through his repentance and prayer will be received by God and inscribed for a good year. Of course he must make some minor adjustments so that the prayer makes sense. For instance, instead of saying "our God and the God of our fathers", he can say "our God and the God of our Patriarchs" since he is not a direct descendant of the Patriarchs. But such obvious points are minor although they do require some awareness of what one is saying. But other than that technical point, all of the prayers, even the piyut (additional prayers) are applicable and beneficial to the Noachide as it is to the Jew. Indeed, the whole theme of Rosh Hashanna is that there is one Creator of the universe and all God's creatures should recognize Him. What then can be more correct before God than to have the Ben Noach recognize Him and pray to Him on this day? Indeed the Ben Noach is in a very special position to do an act that has a special value, a dimension which his unique position allows him to accomplish, which the Jew cannot. As it says in the prayers, "Let all those who dwell on the Earth recognize and know that [only] to You shall every knee bow down...and all shall accept the yoke of Thy kingdom..and God shall be the king of the entire Earth and He and His name shall be one." The Ben Noach prayer has a very special place before God as part of the fulfillment of His words. May the one who asked the question be blessed with all of God's blessings for a wonderful and a spiritually fulfilling year. There is one last point. In order not to violate making a holiday for himself since he is not commanded, the Ben Noach should not treat the day as a holiday by making it festive or imposing any prohibitions on himself. ■