If a person, for whatever reason, did not recite Musaf on Shabbat, Rosh Hodesh or Yom Tob, and the day ended, he cannot make up the missed prayer. When it comes to Musaf, we apply the rule of "Abar Zemano Batel Korbano" – "Once the time has passed, the sacrifice is void." Once the day has passed, the Musaf prayer – which commemorates the Musaf sacrifice – can no longer be recited.
If a person forgot to recite Shaharit but recited Musaf – such as if he arrived in the synagogue late and joined them for Musaf, forgetting to recite Shaharit – he may still recite Shaharit. As long as the final time for Shaharit has not passed, he may recite the Shaharit prayer, even though he has already recited Musaf. Of course, Shaharit should be recited before Musaf, but if one mistakenly recited Musaf before Shaharit he may still recite Shaharit, as long as the final time for Shaharit has not yet passed.
If a person did not recite Shaharit or Musaf until Minha time, such as if one was ill and did not get out of bed until the afternoon, he should first recite Minha, the prayer that is required at that time of day. He should then recite Musaf, followed by the Tashlumin (make-up prayer) for Shaharit. This is the ruling of the Hida (Rav Haim Yosef David Azulai, 1724-1806) and the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909).
Summary: One who missed a Musaf prayer cannot make up the prayer once the day has passed. One who mistakenly recited Musaf before Shaharit may still recite Shaharit. One who did not pray Shaharit or Musaf until the afternoon should recite Minha, followed by Musaf, and then the make-up prayer for Shaharit.
