Arriving at HKIA at around 1am and need to get to Tsim Sha Tsui. Will the mtr still be open?

Well, according to your arrival time, I'm afraid that there is no MTR available.
Thus you are advised to take bus no.N21 at Passenger Terminal Building, Cheong Tat Road, and get off at Middle Road, Nathan Road. It takes around 80 mins. And the ticket fare is 23 HKD per person. After that, walk north along Nathan Road around 250 yards. By the way, the operating hours of this bus line are from 00:20 to 04:40 at intervals of around 20 mins.