When you arrive at Rome-FCO, the train station at the airport is in front of Terminal 3. Simply follow the signs when you get close. You can buy your tickets straight through to Manarola at this station. You should received 3 different tickets. 1) a 14Euro ticket for a non-stop ride on the Leonardo Express to Rome-Termini. 2) A ticket from Rome-Termini to La Spezia Centrale via Civitavecchia and not Florence. This ticket will have seat reservations so I suggest about 30 minutes separation at Rome-Termini as the LE train arrives on track 25 or so and you have to walk quickly into the main terminal to look onto the signboard and find your train and track. Then you have to board. 30 minutes. 3) A Regionale ticket from La Spezia Centrale to Manarola. This ticket is good for 60 days when purchased in Italy. Thus, your reserved train could be late and you simply take the next Regionale train to Manarola. No problems here. It's the reserved seat train from Rome to La Spezia for which you have to plan carefully. No seat reservations permitted on the other two trains. Just get on.
Note that English is often not the agent's 1st or 2nd language (Italian and German with some French and English). Thus, I suggest that you carefully look at www.trenitalia.com and selected the train run you want from Rome to La Spezia plus a second backup run and write them down. Your plane could be late. Show that to the agent at the FCO train station. The Leonardo Express runs every 30 minutes so that's not a problem.
Thank you for identifying your arrival date and time. Sooo Helpful. So helpful in fact that here's a link to a picture of what I see on www.trenitalia.com on April 9 in the late morning and early afternoon. These two train runs fit nicely into your arrival schedule. http://gyazo.com/bc40e801688ed91f6fdcb0a47c79cb71