The fiscal year in the Beit Hamikdash starts on Rosh Chodesh Nissan. The public sacrifices, such as Tamid and Musaf, were to be purchased from the new Machtzit Hashekel collection.
In order to properly prepare, Bet Din would make public announcement on Rosh Chodesh Adar to remind people to donate.
Reading Parshat Shkalim on, or prior to Rosh Chodesh would be part of that preparation.
The Mitzvah of Machtzit Hashekel is limited to the Temple period, and our reading remains as ונשלמה פרים שפתינו, a reminder and longing for the opportunity to fullfil the Mitzvah. The Gemara (Megillah 29) tells us that the month of preparation is based on
שואלין בהלכות הפסח קודם לפסח שלושים יום.
The Poskim (Orach Chaim 429) discuss whether this preparatory month is limited to Pesach, or applies to other chagim as well.
Its application to Machtzit Hashekel is interesting. Is it just a borrowed idea, or is there a deeper connection?
In any case we see that proper preparation is a crucial part of our holidays, Mikdash and Mitzvot.
Shabbat Shalom