Can I take a 21" roll-on bag and backpack as carry-on luggage Air Lingus?
Probably not but it will depend on strict the gate personal are.
The AL website says that carryon is limited to one 22" (including wheels, handle) bag, weighing no more than 22 lbs. You may also have one small personal item, such as a purse. I doubt that they would accept a backpack, unless it was a small daypack type. They are strict about weight.
A daypack the size of a purse, slung over one shoulder, would probably be fine. A backpack that looks like a second piece of luggage won't be.