Can you get from Nice Airport to near Gare de Nice Ville around midnight by public transport?

In general, you can always take a taxi. Taxis are around at the airport as long as airplanes are operating. There is a page in english made by the airport regarding the taxi transfer and the night fares. You can also call a taxi company like this here to pick you up in case there is none at the airport by chance. If you want to go 100% sure that you will get a reserved pickup, you can book a kind of limo/taxi-transfer on pages like this here.

However, there are also options by night bus ("Noctambus"). The issue here is that the city night buses do not stop at the airport and the regional ones do not go to the train station. Also, the Noctambus is for people leaving the city, so it gets more complicated going to the trainstation downtown. I will describe it here for completeness but as you will see this is quite complicated and a taxi will be much easier. So there are 2 ways to do this via bus:

1. Take a taxi or walk to the next station for city night buses, then walk or take a taxi.
2. Take the regional night bus and then walk or try to find a taxi.

This is the map for the all night buses. On the bottom left of the map you have the airport. You can see that the yellow line (N4) stops in the east (Gare SNCF St. Augustin) and in the north (March Gare). According to the N4 schedule, this would be at 0:50 and 0:57 respectively. This one will bring you downtown, but it does not stop at the Gare Nice ville either, since all night buses does so only on the way leaving the city, not when they come back. You will have to walk or find a taxi instead once you are downtown.

If you take the regional bus, (The grey N100 line on the map), you can board it at 1:00 at Terminal 1, and get off at Gambetta-Promenade or Le Port and walk from there or take a taxi if you get one.