Can A Charged Capacitor Charge A Battery?

Yes, a charged capacitor can charge a battery.

To charge a battery via a capacitor, all that is required is a voltage across the capacitor that is greater than the battery voltage.
The
size only affects the time it takes for the capacitor to charge the battery¹.

However, keep in mind that the amount of energy stored in capacitors is usually less than that in batteries, so it doesn’t charge the battery much.