What are the different options to get to downtown Palermo from Palermo Airport?
Rental car-€€€€,taxi,€€€, bus(hourly) €10.
General info:
https://www.airportsineurope.com/airports/palermo-airport/
Train
http://www.trenitalia.com/tcom-en/Services/Palermo-Airport
Bus
http://www.palermo-airport.com/transportation.html
The above are the cheapest options...then there are transfers and then taxis. I would make the decision once you're on the ground. It's not hard to buy a train or bus ticket on the spot.
Agnes's approach is mine when I land somewhere that offers both train and bus transportation into town. I check on schedules beforehand, and I look at my watch as I exit Customs. Then I decide, taking into account cost, where I'll be dropped off in the city and expected waiting time for each option. It's not perfect since buses can be delayed by traffic, but it has always worked out fine.
Check with the hotel and the hotel's web site. Generally that info is posted since it is frequent question. With two people the marginal taxi cost over a couple of bus or train tickets maybe be small. Door to door service with luggage is convenient.
The bus from the airport goes all the way to the train station, which is in walking distance of many of the hotels in Rick's book. I think the idea of checking with your hotel is a good one. Your decision may also depend on how much luggage you have. The sidewalk heading away from the train station is crowded, and cobbledy. Not great for carrying heavy bags or tippy roller bags. Not a problem for backpacks.