Raincoat

Nike maybe call see if they have anything first. Or have your hotel call or ask a concierge at any hotel.
Looks like this Nike store its still open for a bit.
Try googling sports store rome italy.

https://www.google.com/search?client=ms-android-americamovil-us&ei=Dv-eWs2hIYuU0gKCiLHQAg&q=sports+store+italy+rome&oq=sports+store+italy+rome&gs_l=mobile-gws-serp.3..0i8i7i30.17350.23795..25289...0....150.1748.19j3..........1..mobile-gws-wiz-serp.......0i71j35i39j0i8i30j0i7i30j0i67.nXI7Sc343a0%3D#istate=lrl:iv&rlimm=7174513472059454218

+39 06 4893 0565

Piazza dei Cinquecento, 00185 Roma RM, Italy

Also any store that carrys North Face should have a waterproof jacket.
I think asking a concierge at any hotel will help you the fastest.

https://www.thelabelfinder.com/rome/the-north-face/shops/IT/28602/3169070

Poncho ?? Aren't there guys on the street selling them the minute it starts raining ?

Your hotel may be kind enough to give you a large garbage bag. Cut a hole in the bottom for your head and holes at the corners for your arms.

Ponchos and cheap umbrellas sold on nearly every street corner when gray clouds start forming... no worries

Migrants selling rainwear may not come as quickly now that the head of the Interior Ministry, in charge of immigration, is Matteo Salvini