The trip from Rome to Capri can be long and complicated: plan for at least 4 hours from when your flight lands, if you are lucky with all your connections.
From Rome Fiumicino, you have to take the Leonardo Express to the main Rome Termini station, then transfer to a different train to Naples. Once in Naples, you have to take a taxi or the metro to the port, and from there the ferry to Capri.
Since there are so many connections, it's hard to plan ahead. If you purchase your train tickets (keeping in mind that the least expensive tickets don't offer reimbursements for missed trains) and ferry tickets ahead of time, you need to make sure you don't miss those connections. Once you are at the Naples train station, you need to check if it makes more sense to go to the Calata Porta di Massa or Molo Beverello port in Naples for the next ferry to Capri, get a taxi or take the metro, buy tickets, find the right pier, and board. Basically, it's stressful and tiring, especially after a long flight!
If your flight arrives in the afternoon, your only option is to book a transfer with a privat speedboat, otherwise you must spend the night in Rome or Naples before embarking for Capri.
With a private transfer including sedan or minivan with driver and luggage transport to the port in Naples, ferry or speedboat crossing, taxi and luggage transport to your hotel on, consider a cost of around €1000 for up to 5 passengers, and a trip time of 3-4 hours.
For small groups of up to 8 people, the cost of a private transfer is not much higher than the combined cost of public transportation.
Getting to Capri isn't that difficult: you just need to hop on ferry. That said, it can end up being a long and stressful day, especially if you don't know Naples well and are arriving after a long flight with numerous bags, or in high season with its crowds and heat. Finding a taxi and the correct pier, waiting in the line at the ticket office, and going through the long boarding process isn't relaxing even for locals, let alone those coming to the island for the first time!
The best option to make the crossing simpler is a private transfer. You can book either a car transfer with ferry crossing, or a car transfer with a speedboat crossing... both directly to Capri.
Private transfers include pick up at the Naples airport or train station, car or minivan transfer to the port, crossing by ferry or private speedboat, and transfer from Capri's port to your hotel. The trip takes about 2 hours.
If you are arriving in the late evening, it's the only way to get to Capri without having to stay the night on the mainland.