Connecting through FCO without Schengen visa

Since you have to buy two tickets, yes, this will be a problem. You will be immigrating into Italy to catch your second flight and that will require that you meet the requirements to be visiting Italy. They’re not going to take your word that you are going to get on that second flight that’s on a separate ticket.

This might be possible on one ticket (I’m not sure) but it’s definitely not worth the risk on a two ticket arrangement

yup, what Carol said.

But he Schengen thing is on a continuously moving period of time 90 days in any 180 day period functionally means 90 days in the LAST / IMMEDIATELY PREVIOUS 180 days (counting the current date). You only need one day in Schengen. If the first of your "In-Schengen" days that adds up to 90 in the last 180 was 160 days prior to maxing out then you get a day back in 21 days. If you did all 90 days at the end of the 180 day period, then, well, you going to have to wait 91 days to get a day back.

But dont listen to me. I screw up a lot but do go check your status with a Schengen calculator on line. See if you have aquired a few days since you maxed out.

Are the gates in the same secure area and can you carry on your bag and check in on your phone for the second flight? It may be possible even with 90 days maxed out.

You need more information about your flights (terminals mainly). If you can be checked in on both, have boarding passes, no checked bags, staying within the same terminal, then technically you can stay airside and be OK. You will need to follow the connecting flights signage to stay airside, there will be a security check.

You may also have to show proof of onward travel, both to the Tirana-Rome airline and to Security if asked.

Any reason that causes you to need to leave airside (Baggage, terminal change not via a secure zone, need to "check in") will ruin it.

If both arriving and departing flights use gates in the non-Schengen area of Terminal 3 - the E gates, and you have carry-on luggage, you SHOULD be able to stay in the E gate area and avoid passport control. But this is theory, not based on experience. And what happens in a worst-case scenario, i.e. your outgoing flight is cancelled or your incoming flight is seriously delayed? So is "cheapest" worth the risk?

https://ec.europa.eu/assets/home/visa-calculator/calculator.htm?lang=en

Enter your dates and see if you are truly 'maxed out'.

If you have no checked baggage you don't need to leave the extra-Schengen area in Fiumicino airport and you don't need to go through passport control.