You need a full Schengen visa (type C).
An Airport Transit Visa (ATV) only allows airside transit.
While you don't have checked luggage and think (*) you can stay airside, for the first airline, your final destination is Italy, so you need to have the relevant paperwork to enter Italy (or the Schengen Area, more generally), so in your case a full visa.
While there are exceptions, you can count on Ryanair to not apply any here. They tend to check paperwork even when it's not needed, so there is no doubt in my mind that they will flatly refuse boarding if you don't have the relevant visa.
(*) Even if you don't have checked luggage, there are many cases where you can't actually stay airside:
- If the airport does not have any sterile transit at all (this is the case in some surprisingly large airports, and especially in airports served predominantly by low-cost carriers)
- If the airport does not have sterile transit for your combination of flights (again, quite often the case when LCCs are involved)
- If you can't get a mobile/online boarding pass for the second flight and need to go to a check-in desk (that would most definitely be true for the same combination of flights in the other direction, no idea about your case)
- In some cases, if your cabin luggage gets checked at the gate due to lack of space or not meeting the allowed limits (especially true in the case of Ryanair)
- And other reasons which may not apply to you like switching terminals or airports, staying overnight, etc.
- Don't forget that flights can be delayed or cancelled, and, especially in the case of a self-transfer, that can lead to circumstances very different from what you planned.