What do Noahides celebrate?
Sukkot.
Can we do that in a Synagogue?
Synagogues are independent, not centralized in a higher authority for all. It depends in what synagogue. Some admit everyone, others only Jews, and others may only admit Jews who are members of that synagogue. Hallachically a Noahide can pray in a Synagogue, omitting those prayers in which Jews state that they are Jewish (since it would be a blessing recited in vain). They cannot be counted in the mynian or read the Torah either. Some Noahides have their own Temples where they pray and study the Torah relevant to them.
Noahides can commemorate all Jewish Feasts as far as they do not rest in the same way as the Jews, since Parashat Noah says that the Covenant with G-d includes the promise that, untill the end of the world, life and work shall not cease on Earth.
However, a Noahide who keeps the Seven Laws is ok, even if he does not commemorate any Feast. They will observe Sukkot when Redemption comes (with a pilgrimage), now it is just voluntary to remember it. Many Noahides try to take their holidays at that time (Just omiiting RITUAL REST).
As for Yom Kippur, a Noahide must not rest or fast, but he may repent from his sins. Other fasts like Tisha Be'Av are not binding on a Noahide, but he is allowed to join the Jews in solidarity with their mourning (don't know if the whole fast is allowed)
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