There are no border controls for travel between the Schengen countries, so you shouldn't have to show your residence permit at all at any permanent routine checkpoints just for flying between two Schengen countries.
The airline will still want to see a passport or other ID at check-in and/or boarding -- but that's not for verifying that you have a right to travel, merely to check that you're the person a ticket was bought for. Airlines do this to maximize their revenue by making sure everyone who wants to travel has to pay them money, rather than buy a ticket off someone else who changed his travel plans. But they shouldn't be concerned with visas or residence permits.
That's in principle, at least. In practice there's a risk of running into an airline agent who has misunderstood what they're supposed to check and does insist on seeing an original visa or residence permit. If that happens, your immediate ways of recourse are very limited. Ryanair is not exactly famous for going above and beyond for customer satisfaction, and since they can probably get away with turning you away (after all you're supposed to have the residence permit with you so you can show it to the police if they randomly stop you in the street), they might do so. You've already paid anyway, and they would save some fuel ...
Still, they might not turn you away (especially when you have a photocopy of the permit), so your Plan A should be to try to travel home on the Ryanair booking as if nothing had happened.
However note that here is an answer claiming that Ryanair in particular has a company policy of checking visas even for intra-Schengen flights. It doesn't say anything specifically about photocopies, so plan A might still be viable.
Plan B: If that fails, buy a train ticket and go overland. There's no airline-style rigmarole when boarding a train, and the conductor won't think he's supposed to play immigration police.
(Afterwards, good luck with getting the Italian authorities to reissue your residence permit!)