Related for Megillah 59:13

ת"ש איזו היא שבת ראשונה כל שחל ר"ח אדר להיות בתוכה ואפילו בערב שבת מאי לאו אפילו בע"ש דומיא דתוכה מה תוכה מקדימין אף ע"ש מקדימין

Come and hear: Which is the first Shabbat [of the series]? The Shabbat in which Rosh Hodesh Adar falls in the previous week, even if it is on Friday. Is it not that “even on Friday” is the same as the middle of the week, so that just as when it falls in the middle of the week we read before, so too when it falls on Friday we read before?

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