Why is Tishrei the only month not included in the custom of announcing the day(s) of Rosh Chodesh on the Shabbos before the month begins?
This article states:
Unlike all other months, Tishrei was not preceded by a kedushas Shabbos -- Shabbos only came into existance after Rosh HaShana/Rosh Chodesh Tisrei. I should have given that idea more thought and explained it better. Putting aside the mysticism, what R' Leibele Eiger means (and it's also in the Shem m'Shmuel) is that what makes, for example, Kislev into Kislev is that fact that it comes after Cheshvan (or to be technical, after the last Shabbos in Cheshvan); what makes Teves into Teves is that it comes after Kislev. The flavor of each month comes from the accumulation of the past that preceded it.
This reason is "drash", but I think it's a nice explanation.
Mishna Berura 417:1
ר"ח וכו' - מנהג קדמונינו לברך את החודש בשבת שלפני ר"ח חוץ מלפני ר"ח תשרי ורמז לזה בכסה ליום חגנו
The custom is to bless the new month on the preceding Shabbos, with the exception of Rosh Chodesh Tishrei. A hint to this is the verse "in hiding the day of our Holiday"
I once heard an explanation that this is in line with the idea of "confusing the Satan" by not giving him a heads up that Tishrei is coming.
R. Reuven Margolios in his sefer Ollelos here explains that when the Sanhedrin used to determine the date of Rosh Chodesh according to the sighting of witnesses of the new moon, it was of course impossible to announce beforehand on which day Rosh Chodesh would fall. But later, when the Romans made decrees which disrupted the announcements of the Sanhedrin concerning the fixing of the calendar, the Rabbis declared that from then on the calendar would be fixed according to their calculations, and thus it became possible to announce to the general population beforehand when the next Rosh Chodesh would be. This they did in the synagogues on the Shabbos before Rosh Chodesh just before Mussaf, since this was the time when most people were gathered together.
However, this announcement was only necessary when the people would have had a doubt as to which day of the week would be the upcoming Rosh Chodesh. This doubt existed almost every month, because even if, for example, they knew which day Rosh Chodesh Nisan had been they still would not know on which day Rosh Chodesh Iyar would fall, because this required knowing whether the month of Nisan that year was 29 or 30 days.
But the Gemora in Beitzah 6a teaches that from the time of Ezra the month of Elul always had 29 days, and thus since everyone already knew which day had been Rosh Chodesh Elul, they all knew for certain which day was to be Rosh Chodesh Tishrei - Rosh Hashanah.
Hence, there had never been a need to announce Rosh Chodesh on the Shabbos before the month of Tishrei, and thus this is still our custom today.