Transferring from a Delta flight to a Ryanair flight at FCO

There is no hard and fast rule. Most say having a three hour cushion between flights that are not booked together should be sufficient, but of course you can always have a long delay.

Not a simple answer. You will need to allow time for immigration and, possible, EES processing. After collecting your checked bags you'll be transferring terminals. Recheck bags with Ryanair. Will you have your boarding pass in hand or will you need to go to the check in desk? Then you'll go through security.

Since you will be traveling on separate tickets, be aware that if your Delta flight is delayed causing you to miss your Ryanair flight, typically you would have to buy a new ticket for the next available flight.

Having flown Ryanair a number of times, even if you think you will "carry-on" avoiding having to check a bag and go back through security, what you bring as a carry-on likely will be oversize and overweight for Ryanair.

Plan on landing, going through immigration, picking up your bag (or not), exiting the secure zone, changing terminals likely, rechecking the bag with Ryanair, proceeding through security, then on to your gate.

Check through the Ryanair information, they have closing times for bag checks, typically 40 minutes before the flight departure. When booking, I would choose the "Plus" option if available, it allows a reserved seat, a 20 kilo checked bag, and airport check-in (in case you have problems with your phone when landing in a new country)

3 hours would be the minimum, 4 would be safer. But any amount is a gamble.

Paul's answer is what I would have said.

One clarification... You can still carry that bag on, you just need to pay. It's cheaper if you pay when you book, more expensive at the gate.

Thanks for all of the replies. You've certainly made it clearer as to what we should do.

I just wanted to add that I have paid to carry on a bag on Ryan air and it is by far the simplest solution. If you have a bag you can carry on Delta, you can pay extra to carry on with Ryan air.