There are some sefarim that bring a minhag not to say "Gut Shabbos" after Mincha Shabbos afternoon because that is the time when Yosef, Moshe Rabbeinu and Dovid HaMelech were niftar, and we behave then like mourners who do not greet people. Presumably this does not apply to one who davens Mincha very early, only close to sunset.
The Rema mentions different minhagim regarding drinking water between Mincha and Maariv after sunset Shabbos afternoon, based on a Medrash that one who drinks water then is stealing from the dead. The concept seems to be that the souls of sinners who passed away are released from the punishment of Gehinnom for the duration of Shabbos and drink towards the conclusion of Shabbos to refresh themselves before they return, and one who drinks then is stealing their water. While this is very difficult to understand, it is brought in Tosafos and many Rishonim in various forms. The conclusion of most Poskim is that this only applies to one drinking water directly from a river, other liquids or water in the house are permitted. The prayer Tzidkascha Tzedek was enacted in place of Tachanun for both of the above reasons, commemorating the passing of Yosef, Moshe Rabbeinu and Dovid HaMelech and also in preparation for the souls returning to Gehinnom.