How can I backup my old data from an Apple II plus (DOS and PRODOS)?

Sorry, not a clue! I will have to sleep on that one. : )

Spiacente, non un indizio! Dovrò dormire su quell'.

Just for information-you won't find an emulator/program that will use the B-side of the 5,25 disk.You will need to copy B-side to A-side on Apple.
You may look for that on google-group/Apple II
Have luck
The easy way is t o copy on3,5" disk PRODOS -that was my second solution.First was with crossworks-DOS PC(ten years old,than new).Crossworks should be cheap(inculding Superserial Card,Program, about 10 Euro-sold it for that.
For modem-like transmission-search for "Jan Hedlund"-advice