Can a charged capacitor charge a battery?

Yes, a charged capacitor can charge a battery.

All you need to charge a battery from a capacitor is to have more voltage charged on the capacitor than the voltage of the battery.

The size will only affect how much time the capacitor will charge the battery¹.

However, it is important to note that the amount of energy stored in capacitors is generally less than in batteries so it wouldn’t charge the battery very much.