I think there is a airport train that runs from Fiumicino to the
central station (Roma Termini).
Yes there is. It's the Leonardo Express mentioned earlier. 14 euro pp.
http://www.trenitalia.com/tcom-en/Services/Fiumicino-Airport
Also do you recommend me take the Leonardo Express Airport Train to
Roma Termini station or the Fiumicino Airport bus? I’m not sure what
is the price and travel time difference?
The L.E. train to Termini takes 32 minutes. Buses (Terravision, SIT, T.A.M., Atral) from the airport to Termini take about an hour and cost between 5 and 6 euro.
http://www.adr.it/en/web/aeroporti-di-roma-en-/pax-fco-bus
High-speed trains from Rome to Salerno: most of us do not recommend pre-purchasing train tickets for arrival day as a delayed or cancelled flight could have you missing your train and losing the cost of the tickets. Italo's FLEX ticket, that Dario mentioned, might be a good buy as it allows you to change your ticket up to three minutes before your orignal departure time at no cost if needed (read the fine print). The only possible snag is that there are few Italo trains to choose from if you miss one.
https://www.italotreno.it/en/the-train/fares
Just guessing that Sunday is your departure from Rome, there's a 12:15 and 14:15 and 20:15 train. if your plane is on time, you should be able to make the 14:15 with no problem but if you miss it, it's a long wait for the 20:15 and would mean driving to your hotel from Sorrento late at night. Also, a car-rental office might not even be open that late.
Otherwise, choose a Trenitalia train (they have more trains running that route) when you arrive in Rome, and be willing to pay for base fare tickets if that's all that's available. At this late in the game, you don't really have a lot of options to play with.