It’s actually according to the Rubrics of the Roman Martyrology, of which the proclamation of the Nativity is part of. The Martyrology (of which I have a copy, in Latin, the most recent editions which BTW I noted for chant), has rubrics for the recitation of the Martyrology as part of the Liturgy of the Hours either after Lauds, or one of the Minor Hours, or for recitation outside the Liturgy of the Hours. However the abbey does the Martyrology reading normally outside the LOTH, as one of the refectory readings before lunch, but they do the Nativity proclamation immediately after Lauds.
Yes, the Martyrology for any given saint or feast is always proclaimed by anticipation the day before.
This is one of those things that would not be specifically Benedictine, but would be in the Customarium of each individual abbey, since the rubrics of the Martyrology themselves allow latitude on when to proclaim the Martyrology (including the Nativity proclamation).
What is specifically Benedictine is the use of the Monastic martyrology, which emphasizes monastic saints and leaves out some minor non-monastic saints. It was published by Solesmes in 1977. Our abbey has recently updated it and re-issued it in its own private edition to include Canadian saints and newly-added saints. I have a copy and it’s what I use. I normally read my commented chapter of the Rule, and the next day’s Martyrology, just prior to Compline.