Does anyone know if public transport will be running over New Years?

It certainly will be, and will be very crowded. It is likely too late now, for anything that requires a reservation. Be prepared for long queues, with no guarantee that you will get a seat on public buses or trains. Go to the station as early as possible in the morning. Everyone has Dec. 31, and Jan. 1, 2 and 3 off, so travel outside that period if you can.