On the Shabbat before any new month, a special prayer called Birkat Ha-ḥodesh is recited in the synagogue to announce the name of the new month and the day on which it is to begin. This is not done in anticipation of Rosh Hashanah, however (Mishnah Berurah to SA Orach Chayyim 417, note 1).
The most logical reason for this omission is that Rosh Hashanah is deemed the beginning of all beginnings, thus the benchmark by which all the other festivals will be measured, and as such is not in need of any sort of formal introduction. Popular folklore, however, has suggested an alternative explanation.
There is a traditional conception that we are all judged during the so-called Ten Days of Repentance that fall between Rosh Hashanah and Yom Kippur.
In this drama, God plays the role of judge and we are the defendants whose very lives are on the line. Satan, playing the role of prosecuting attorney, anxiously awaits the arrival of Rosh Hashanah every year so the trial can begin and he can present all the most damning evidence he can gather against each defendant.
Accordingly, we do not announce the coming of Rosh Hashanah in public in the hope that omitting the prayer will somehow confuse Satan, thus causing him to miss the trial entirely and leaving God to judge us mercifully.
This same explanation is given as to why we do not blow the shofar on the morning before Rosh Hashanah, although the far more likely explanation is the one given above: that we omit it to distinguish between the blowing of the shofar during Elul, which is a custom, and the blowing of the shofar on Rosh Hashanah, which is a mitzvah.
Adapted with permission from The Observant Life.