My usual response to posts about going to Sicily for 4 or 5 days is "Please don't", but you have a family connection, so I can understand why you want to squeeze it in. You can probably accomplish a good bit with four nights there, but I hope you'll consider being satisfied with a view of Mt. Etna from afar. If you want to walk up the cone, I think you'll find it takes up a good bit of the day. Mt. Etna is north of Catania (and visible from that city if the weather is clear), whereas Siracusa is to the south.
The company Azienda Siciliana Trasporti runs buses between Siracusa and Ferla (it's Line 788), but only once a day (I just checked a weekday). So that probably isn't what you want to do. The distance from Siracusa is over 25 miles. Renting a car in Siracusa would be a more economical option than a taxi or car/driver arrangement, I assume, though I've never rented a car in Europe. I didn't observe anything that made me think rural driving in Sicily was a problem, other than some narrow streets when roads wound through towns.
If you are uncomfortable about driving there, you could certainly arrange for a driver in Siracusa. This probably wouldn't be exorbitant if you just want the driver to take you there, wait while you look around for a little while, then drive you back. I don't know anything about Ferla, so I don't know how much time you'll want to be there. The VisitSicily website to which I linked suggests that it would be worth spending some time there. Certainly if you want to do some genealogical poking around while you're there, keeping a taxi waiting might get pretty costly.
There are some other lovely villages in that area west of Siracusa that I visited briefly in 2015. All can be reached by train or bus, and they have frequent enough service that public transportation is a viable option (though not on Sunday). Siracusa has a lot of attractions of its own, but if you have extra time, I highly recommend seeing some or all of Ragusa, Noto, Modica and Scicli. Ragusa is the one that takes the most time. A rental car would allow you to cover a lot of this ground fairly quickly, and still leave time for Siracusa itself.
