Taxi costs and duration

I need to be at Pudong airport at 5:45/6:00 in the morning. I will start at Mingzhong Road (Songjiang district).
I thought about leaving at 4:45. The taxi fare at night is until 5, so do I only need to pay 15min of the night fare and the rest day fare or does the night fare count the whole drive because I entered the taxi before 5?
And how long will the drive take? Is 4:45 okay or too late?