Pompeii for a day

" We want to catch an early evening train to Rome FCO"

So, you really don't have a full day- only about 9 hours. Yes, it can be done, with about 4 hours to look around. I wouldn't do it, but it's your trip & your decision.

Yes, you can use the baggage deposit office at Napoli Centrale station near track 5, €5/bag for the first 5 hours, then .70/hour. The baggage room is only staffed until about 20:00 (confirm when there). The last departures of the fastest (1-hour) trains to Roma Termini are about 19:00 and 19:30 (but earlier on Saturdays). From Naples to Roma Fiumincino, trains take 2.5 - 3 hours with a connection at Termini. Travel time on the Circumvesuviana to/from Pompei Scavi is about 40 minutes each way. If you decide to explore Naples instead of Pompei, then don't miss the Archeology Musuem.

If you're taking an international flight from FCO you need to be at the airport three hours before departure. It will be a very tight day and very tiring. The fares from Naples to Pompeii are 4.20 euro one way and entry to Pompeii is 11 euro per person. Join a group tour as the audio guides are out of date and a waste of money. A group tour will last about two hours and take you to all the major places inside.
You should reserve your Napoli Centrale - FCO tickets in advance to secure a cheap fare as well as being guaranteed a seat. Trenitalia are expanding their fast services all the way to FCO but I am unsure when this starts, so you may need to change trains at Roma Termini for the Leonardo Express to the airport. The Leonardo only runs every half hour. You will need to factor this in to make your flight.
Be aware of unexpected delays. I wouldn't risk doing what you have planned on the day of an international flight but if you're going somewhere else in Europe, it is slightly less of a risk.

This should be totally doable in the time you list. Given that you are a bit time constrained I would suggest looking into a private or small group tour of Pompeii if you can. There are a ton of great ones out there depending on your budget and likes and an online search should turn up more than enough options to chose from. Tours will keep you from wasting precious time you don't have getting lost/looking for things. Furthermore, a lot of these tours allow you to skip waits/lines for things which will give you more time to experience the sites vs. waiting in line.

As far as leaving bags and traveling my recommendation is to do everything you can to get on the 6:30 (ish) boat from Ischia to Naples vs trying to leave from Forio. Then go to Napoli Centrale and drop off your bags before taking the Circumvesuviana train up to Pompeii. These leave every hour or so and take about 45 minutes. Then upon your return to centrale via Circumvesuviana train both the Italo and Trenitalia trains heading to Rome leave from that station so it should be easy. That said the Italo train leaves you in Rome's Tiburtina station vs Trenitalia which leaves you in Termini so make sure you are aware of that so you don't mess up travel on the Rome end of things.

Something to note, the Circumvesuviana trains are not always the most on time, something which can pretty much be said for most trains in Italy (and let's not talk about France). Therefore, I would leave a good amount of buffer time and not cut things too close. Otherwise you're asking for pain and stress.

Hope this helps and enjoy your trip!

Thanks very much for the helpful suggestions!