All the cell phones we have are charged using a USB standard cable and connector. They can be charged using a laptop computer USB connection or an AC adapter with a USB type cable. The cell phoned require 5 volts DC through the USB port to charge the battery. All charge controlling is inside the cell phone.
All you need to do is supply 5 volts DC at, say, 0.5 amp. I don't see why an Arduino needs to get involved.
Paul