The Shulhan Aruch (Orah Haim 114:2) writes that one may not begin reciting "Mashib Ha’ruah" in the Amida prayer until the Hazan announces before Musaf on Shemini Aseret that it should be recited. Therefore, if a person prays at home, such as if he is ill or for some other reason cannot attend the prayers in the synagogue, he should not pray Musaf until the time the congregation prays. He must wait until the time when he can be certain that the Hazan had made the announcement to begin reciting "Mashib Ha’ruah." At that point, even though he did not hear the Hazan make the announcement, he may recite Musaf with "Mashib Ha’ruah," since the announcement was made in the synagogue. Likewise, if a person comes to the synagogue late on Shemini Aseret and sees that the congregation had begun reciting Musaf, he may recite the prayer at that point with "Mashib Ha’ruah," even though he did not hear the Hazan’s announcement.
Hacham David Yosef, in his Halacha Berura, discusses the case of a person who finds himself in a place without a Minyan on Shemini Aseret. He writes that in this case, one must wait until the time when most congregations have begun reciting Musaf, which is after six hours into the day.
Hacham David also addresses the situation of a place with multiple Minyanim, such as if one lives in a city where some people pray very early (at "Netz") while others pray later (as is the case in Brooklyn and most large Orthodox Jewish communities). He writes that if a person prays at home, he may begin Musaf after the time when the early Minyan begins Musaf, even if he does not normally pray at the early Minyan. However, if one is praying in the later Minyan, and wants to pray Musaf on his own before the Minyan reaches Musaf, it is uncertain whether he may rely on the announcement made by the Hazan in the early Minyan.
Finally, Hacham David writes that if a person arrives at the synagogue on Shemini Aseret while the congregation recites Musaf, and he begins praying Shaharit at that point, he does not recite "Mashib Ha’ruah" in Shaharit. Even though he is covered by the announcement made in the synagogue, he begins reciting "Mashib Ha’ruah" only during Musaf, and not during Shaharit.
Summary: One may not begin reciting "Mashib Ha’ruah" on Shemini Aseret until the Hazan in the synagogue announces before Musaf that it should be recited. Therefore, one who prays at home on Shemini Aseret should not recite Musaf until the time he expects the first Minyan in the town to begin reciting Musaf. If he is in a place without a Minyan, he should not begin Musaf until the end of six hours into the day. If a person arrives in the synagogue after the congregation began reciting Musaf, and he begins praying Shaharit, he does not recite "Mashib Ha’ruah" in Shaharit but does recite it in Musaf.