What are some ways and prices for getting from the airport in Palermo Sicily to an Airbnb?

Look up the shuttle bus to Centro Palermo on the airport's website - probably, no other option will be less expensive unless you can split the taxi costs with X number of friends (not sure what a taxi would cost since you haven't provided any info on where your destination airbnb is).

You may get more specific information if you tell us where the Airbnb is.

Hi neighbor! We took a taxi (three of us) as door to door transportation is important to us when we are jet lagged and, being three, cost is less of an issue. It was 38 Euros to central Palermo.

Ask your landlord?

I am staying at this address!
Vicolo delle Api, 7, Palermo, Sicilia

Taxi seems to be your best bet.

https://www.rome2rio.com/map/Palermo-Airport-PMO/Vicolo-delle-Api-7-90134-Palermo-PA-Italy

Click on the options for more info.

Google "Palermo Car Service TripAdvisor", for example, Alex Travel is one that is highly rated.

Train from PUNTA RAISI to PALERMO NOTARBARTOLO. Cost is €2-6 per person.
Walk 1 km. (Or take a taxi for €9.)
That is what we plan to do.