If one is davening at home on the first day of Shemini Atzeres and he has not davened Shacharis yet, but the tzibbur has already davened Mussaf and said mashiv haruach, should he insert mashiv haruach in his Shacharis davening?
Shulchan Aruch, Orach Chaim 114:1 clearly states that Mashiv HaRuach is inserted beginning with Mussaf. He does not differentiate between an individual or a community.
מַתְחִילִין לוֹמַר בִּבְרָכָה שְׁנִיָּה: מַשִּׁיב הָרוּחַ וּמוֹרִיד הַגֶּשֶׁם, בִּתְפִלַּת מוּסָף שֶׁל יוֹם טוֹב הָאַחֲרוֹן שֶׁל חַג, וְאֵין פּוֹסְקִין עַד תְּפִלַּת מוּסָף שֶׁל יוֹם טוֹב הָרִאשׁוֹן שֶׁל פֶּסַח.
We begin saying in the second bracha [of Shemoneh Esrei] Mashiv HaRuach u'Morid HaGeshem in Mussaf of Shmini Atzeres, and we do not discontinue until Mussaf of the first Yom Tov of Pesach.
In Halacha 2, however, he discusses one who is unable to get to shul:
אָסוּר לְהַזְכִּיר הַגֶּשֶׁם עַד שֶׁיַּכְרִיז שְּׁלִיחַ צִבּוּר הגה וְיֵשׁ אוֹמְרִים שֶׁקֹּדֶם שֶׁמַּתְחִילִין מוּסָף מַכְרִיז הַשַּׁמָּשׁ מַשִּׁיב הָרוּחַ וְכו', כְּדֵי שֶׁהַצִּבּוּר יַזְכִּירוּ בִּתְפִלָּתָן, וְכֵן נוֹהֲגִין, (מָרְדְּכַי רפ''ק דְּתַעֲנִית). הִלְכָּךְ אַף אִם הוּא חוֹלֶה אוֹ אָנוּס, לֹא יַקְדִּים תְּפִלָּתוֹ לִתְפִלַּת הַצִּבּוּר, לְפִי שֶׁאָסוּר לְהַזְכִּיר עַד שֶׁיֹּאמַר שְׁלִיחַ צִבּוּר; אֲבָל אִם יוֹדֵעַ שֶׁהִכְרִיז שְׁלִיחַ צִבּוּר, אַף עַל פִּי שֶׁהוּא לֹא שָׁמַע, מַזְכִּיר
It is forbidden to mention the rains until the Chazzan announces. Rema: Some say that before they begin Mussaf, the Gabbai announces Mashiv HaRuach so that the community may mention them in their prayers, and thus is the custom. Therefore, if he is sick, or forced [to be away from minyan], he shouldn't precede his davening to that of the community, as it is forbidden to mention until the chazzan says it, but if he knows that the chazzan has announced, even though he did not hear [the announcement], he may mention.
The Mechaber seems to be addressing Mussaf; in fact, Mishnah Berurah and Be'er Heitiv to Halacha 2 seem to indicate as such (though I wouldn't say that with certainty - I will rescind that claim if someone can bring an inference to the contrary). Having left Shacharis unaddressed, it would seem that the Mechaber holds that it reverts to the ruling in Halacha 1, that we do not say it in Shacharis.
Mishnah Berurah 114:2 notes, though, that if one did say it he need not return to the beginning of the beracha, as he would normally (as per OC 114:4), since by right we should be saying Mashiv HaRuach from Maariv the night before (see there and Be’er Heitiv OC 114:1 for reasons why we don’t).