There is no authentic report from the prophet predicting the martyrdom of the Hazrat Husain.
The fast of 10 Muharram was a Nafal fast which the prophet told his companions to keep. Ghamidi saheb writes in his book Meezan:
The Fast of ‘Ashur (10th of Muharram)
“Ahadith mention the blessings of this fast. The Prophet (sws) generally kept this fast and before the fasts of Ramadan were made incumbent, he would necessarily keep it and would urge and direct people to keep it and would be vigilant on them in this regard. According to history, one of the reasons for which this fast was kept was that the Quraysh used to keep it and another reason recorded is that the Jews would keep it. When the Prophet (sws) asked the Jews, they replied: “This day has great significance for us; the Almighty liberated Moses (sws) and his people on this day and drowned the Pharaoh and his people in the sea; it is to express gratitude to the Almighty that Moses (sws) fasted on that day.” At this, the Prophet (sws) said: “We have deeper relations with Moses (sws) than you.” Consequently, he fasted on this day and also asked people to fast.” (Objectives of Fast)