Taxis are usually available at the port as soon as a ship docks; taxi drivers will know, if a ship arrives for disembarkation on top of it.
There's an information desk at the tiny port building with a hostess, who will be able to help you, if the few taxis are already taken (Monaco does not have too many cabs; it's a small city). Also, people on the ship have that information, too, and sometimes even reserve taxis ahead of the arrival - all depending, which cruiseline you take.
If you find no assistance at all (which would be strange), here is the direct number of the taxi service: 08 20 20 98 98.
Taxi fares to Nice should be something between 70 and 80 Euros. They are obliged to use their meters.
If you find other guests in the ship heading for Nice, too, I suggest to hire a private transfer for 4 or 6 people, which is only slightly more expensive, but maybe easier, since the driver will be waiting for you at the requested time. You will get information at: email@example.com
Hope you will have the chance to explore a bit of Nice and Monaco as well, when you disembark in the beautiful South of France :-) .