Your question is asked in a Sefer published two years ago in Yerushalayim by Rav Dovid Cohen. The name of the Sefer is Zman Simchaseinu, and the subject is Sukos and Shemini Atzeres. On page four he asks your question.

1. Rav Cohen answers that the Gra, on Seder Olam Rabah, disagrees with Tosfos. The Gra writes there that the 9th of Tishrei of that year fell on Monday, and on that day Hash-m forgave the Jewish people. However, Moshe Rabeinu did not yet come down from Har Sinai because the 40 days had not yet been completed. Since, according to this, Yom Kippur fell on Tuesday, it follows that the 15th of Tishrei was on Sunday, so they were able then to start building the Mishkan. The Gra therefore is consistent with his own view that the construction of the Mishkan commenced on the 15th of Tishrei, because if Yom Kippur of that year would have fallen on Monday it would not have been permitted to build the Mishkan on the 15th of Tishrei which falls on Shabbos. Rav Cohen adds that the Ba'al ha'Turim (on Shemos 36:6) also writes that Yom Kippur of that year fell on Tuesday.

2. However, Rav Cohen cites another source in the name of the Gra (from the compedium Torah mi'Tziyon cited in Likutei ha'Gra, Mo'adim) that the first day of Sukos of that year fell on Shabbos. This agrees with Tosfos in Bava Kama (82a) and contradicts what the Gra writes on Seder Olam. According to this, it indeed is difficult how the Mishkan could have been built on Shabbos, but again if we follow the Gra in Seder Olam there is no difficulty.

Kol Tuv,

Dovid Bloom

1. The Vilna Ga'on on Shir ha'Shirim 1:4 writes that on the 14th of Tishrei all the wisehearted men in Klal Yisrael took the gold from Moshe Rabeinu according to its amount and its weight. It appears, then, that the prohibition involved in doing this on Shabbos is that of Muktzeh -- moving the valuable gold. If so, one can explain that this was permitted, according to the Vilna Ga'on, because one is allowed to move Muktzeh if it is necessary for performing a Mitzvah. This is stated in the Bi'ur ha'Gra to Shulchan Aruch Orach Chaim 586:22: "d'Mukzteh Mutar b'Mitzvah" -- the prohibition of Muktzeh is waived if moving the Muktzeh enables one to do a Mitzvah.

2. Of course, one could give a simpler answer than the above and assert that by the time of Moshe Rabeinu the rabbinical prohibition of Muktzeh had not yet been instituted.

3. Yes, there does seem to be a contradiction in the Gra's words. In fact, with the help of Reb Meir Bergman's comment, I can add another apparent contradiction. The source for the reason mentioned above for the long Tachanun on Monday and Thursday is the Tur (Orach Chaim 134). The Gra himself on the Shulchan Aruch there cites Tosfos in Bava Kama (82a) as a source for the fact that Moshe Rabeinu came down on a Monday. We see that the Gra himself cites Tosfos even though the Gra disagrees with Tosfos. This contradiction clearly requires further study.

Here is an explanation for the apparent contradiction in the words of the Vilna Ga'on.

1. We often find "Midrashim Chalukim" -- different Midrashim which disagree with each other. The fact that the Midrashim are not in agreement is not problematic, because we know that there are "70 faces to the Torah."

2. I will attempt to provide another proof that we must use this answer of "Midrashim Chalukim" in response to the question concerning when Moshe Rabeinu descended from Har Sinai. This again is prompted by Reb Meir Bergman's comment about the long Tachanun on Mondays and Thursdays. We find that the Tur (Orach Chayim 134) explains that those days are days of favor from Hash-m because Moshe ascended on Thursday and descended on Monday. However, we saw in our first reply to the original question that the Ba'al ha'Turim (Shemos 36:6) writes that Yom Kippur of that year fell on Tuesday. Of course, the Ba'al ha'Turim and the Tur were written by the same author, and thus we must now say that there is also a contradiction in the words of the Tur!

3. The answer is that in fact there are Midrashim Chalukim which long precede the Tur and the G'ra. The Roke'ach (#209) cites the Midrash Tanchuma that states that Moshe Rabeinu descended on a Monday (this is cited in the Masores ha'Shas to Tosfos in Bava Kama 82a). However, as we have seen, the Seder Olam states that Moshe Rabeinu came down on Tuesday. This whole question, then, is actually a dispute between the Midrash Tanchuma and the Seder Olam.

4. Consequently, it seems that neither the Tur nor the G'ra wanted to decide a dispute in Chazal, and they merely cited both opinions. (See Ra'avyah, part 3, Hilchos Ta'anis #849, DH Higi'ah Shiv'ah, who reconciles the two Midrashim.)

However, the G'ra wished to give his new explanation about why we make Sukos in Tishrei rather than in Nisan. Since one of the reasons for this is that Bnei Yisrael began to build the Mishkan on 15 Tishrei, it was necessary for the G'ra to rely on the Seder Olam that Yom Kippur was on Tuesday. This does not mean that he rejected the other Midrash that Moshe Rabeinu descended on Monday.

The Vilna Ga'on, in his commentary to Shir ha'Shirim (1:4), writes that on the 14th of Tishrei every wisehearted man -- who would later skillfully build the Mishkan -- took the gold from Moshe Rabeinu according to the apporopriate amount and weight required to build the Mishkan. After doing this they were ready to start building the Mishkan on the 15th. Since there does not appear to be any Torah prohibition involved in taking the gold on Shabbos, we now understand how they were allowed to do this on the 14th.

Kol Tuv,

Dovid Bloom