And why on the first of Adar? So that Israel may bring their shekels in their proper season, and the contribution of the chamber may be set aside from the new [shekels] at its appointed time, on the first of Nisan. Rabbi Yitzchak said: At the outset, "And it came to pass in the first month" (Exodus 40:17), and it is taught concerning it: on the day that the Tabernacle was erected, on that very day the contribution was set aside.
Why the Shekels Are Gathered in Adar Before Nisan
Yalkut Shimoni on Torah 386:10
וְלָמָּה בְּאֶחָד בַּאֲדָר, כְּדֵי שֶׁיָּבִיאוּ יִשְׂרָאֵל שִׁקְלֵיהֶן בְּעוֹנָתָן וְתוֹרְמִין תְּרוּמַת הַלִּשְׁכָּה מִן הַחֲדָשָׁה בִּזְמַנָּהּ בְּאֶחָד בְּנִיסָן. אָמַר רַבִּי יִצְחָק, בַּתְּחִלָּה (שמות מ, יז) "וַיְּהִי בַּחֹדֶשׁ הָרִאשׁוֹן", וְתָנִי עֲלָהּ, בַּיּוֹם שֶׁהוּקַם הַמִּשְׁכָּן בּוֹ בַּיּוֹם נִתְרְמָה תְּרוּמָה.