Private driver Pisa airport to apartment in Florence

Here's one based nearby: https://www.noleggioconconducente.it/ncc-aeroporto/ncc-aeroporti-toscana/aeroporto-di-pisa-psa-galileo-galilei.html

Also Viator acts as a broker for drivers -no surprise there.

It might help to know there is easy bus service from the Pisa airport to Florence: https://www.pisa-airport.com/en/the-passengers/transport/bus.html

or I expect that the new people mover gizmo is done by now which feeds the Pisa station and really easy train from there to Florence (SMN) which is in the middle of town.

The people mover at Pisa airport is indeed in service, FYI, making train connections a bit easier than they used to be. Expensive though.

If you tell us approximately where your apartment is in Florence, we can give you help to get there from the train station....if you take the "people mover" from Pisa Airport to Pisa Centrale, then train from Pisa Centrale to Florence SMN station.

The private transfer is probably somewhat expensive since the distance is over 80km (50 miles).
I suggest the bus option, as mentioned above, especially if you have luggage, because the bus driver would put your luggage in the bus belly and from the Pisa airport to Florence would require no changes. You just sit comfortably in the luxury bus for 70 min till you arrive at destination. I think the bus drops you off at the Fortezza now (no longer at the train station, but not too far from it). From the bus drop off you can take a taxi to the apartment.
The Pisa Mover from airport to Pisa Centrale station + the Regionale train to Florence is another option, but not as convenient in my opinion, because you need to carry your luggage over 2 different modes of transportation (people mover then train). Also you are responsible for carrying your luggage aboard the train and onto the the overhead compartment above your seat, that is why the bus is a better option if you have substantial luggage.
The time of travel is similar. The bus takes 70 min. from in front of the airport to the Fortezza in Florence. The people mover+train combination, once considered the time necessary to walk to the people mover station and from the people mover to the train tracks is at least as long as the bus.