The Noachide should know that he too is judged on Rosh Hashanah by G-d 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 G-d and through his repentance and prayer will be received by G-d 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 G-d and the G-d of our fathers", he can say "our G-d and the G-d 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 Hashanah is that there is one Creator of the universe and all G-d's creatures should recognize Him. What then can be more correct before G-d 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 G-d 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 G-d as part of the fulfillment of His words. May the one who asked the question be blessed with all of G-d'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.
[By Rabbi Yisroel Chait, the head of Yeshiva B'nei Torah in Far Rockaway, NY, reprinted from: Noahides and the NewYear]