1) How do i explain to my shia classmates that its good to fast in Muharram?

2) And how is tahajjud prayer prayed? The one at 12am. I want to know how many raka's and what surah to recite?

1) Fasting on 1st day of Muharram (Al-Muraqibat – Mirza Tabrisi) is like spending the entire year in good actions; there is protection until following year and if one dies within the year, he/she secures a place in Janna. In Nafasul Mahamoon (Shaykh Abbas Qummi) quotes: Imam Ridha (pbuh) asks his companion, Ar Rayyan ibn Shabeeb – “Are you fasting?” He answers no. Imam then says: “Yabna Shabeeb! This is the day when Zakariyya at 75 years prayed to his Rabb (3:38) for a child. Allah accepts his dua and commands angel to go to his mihraab and announce the good news of a son Yahya (3:39). One who fasts and invokes Allah He will accept as the dua of Zakariyya”
It is mustahab to fast on the first nine days of the month.

2) The time for praying Salatul-Layl is from midnight (midnight is reckoned as the time which is exactly half the time between sunset and the actual dawn) to the time of Adhan for Fajr prayers. However, it can be prayed at anytime after Isha prayers but it is better to pray during the last part of the night near to the time of Fajr prayers. If it is not possible to pray after midnight or before Fajr, then it can be recited in the morning or any part of the day with the niyyah (intention) of Qadha. It is better to recite with the niyyah of Qadha during the day time than to pray in the early part of the night. 

Salatul-Layl is made up of Eleven Rak’ats.