Nowadays is an individual allowed to accept a fast upon himself at Mincha before the next day? I heard that nobody nowadays does it.

Why not? Shulchan Aruch Orach Chayim 562:6 describes how to accept upon oneself a Ta'anit in Mincha of the day before. More so, while saying Hatarat Nedarim on Erev Rosh Hashana most minhagim add a sentence explicitly allowing one to accept a Ta'anit. Rabbi Ro'i Margalit