Destinations from Rome airport

You could fly to Naples and take the train from there. Or you could drive, which would be discouraged if coming off a transatlantic flight. Trains or buses would rrequire a change in Rome.

If you mean driving, yes. You take the freeway east toward Rome then the GRA (Grand Circular Beltway, which bypasses the city center) counterclockwise to the direction of Naples.

By train you need to change trains at Roma Termini.

Perhaps someone else knows of such a possibility via public transportation, but I can't find anything by Googling. It's not really difficult to take the Leonardo Express from the airport (Fiumicino Aeroporto) to Roma Termini station and switch to one of the fast Freccia trains there. Roma Termini is a dead-end station, so upon arrival you walk to the head of your train and simply walk along to the left or right to get to your new platform. You can use the app to check on the new platform before you step off the first train, or you can look at the electronic signage after arrival. You should double check the signage at the head of the new platform in case there has been a last-minute change.

There are multiple connections each day that will get you to Salerno in less than 3 hours, plus others that take between 3 and 4 hours. Some departures require a second change in Naples (Napoli Centrale), but many do not.

Although some airports have bus transfers available to other cities, I can't find any such option from Roma Fiumicino to Salerno--quite possibly because the trains from Roma Termini to Salerno are very fast.

You could of course get a taxi all the way to Salerno or arrange a private transfer in advance, but the distance is about 170 miles each way, and you'd be charged for the riderless trip back to Rome, so it would be a very costly experience.

Yes

One example
https://www.vip-charter-service.com/Italy/helicopter-transfer/Salerno/index.html

Thanks for your help.