I believe desulfating is based on trickle charging, so doing it via just solar might work. And if we're talking apocalypse, it'd be good to have a secondary source of power like something that can run off of organic material such as wood gas.
The desulfator i speak about is powered by the 12V battery itself. It's a small box with just get hooked up to positive and negative. Technology differs but basically it charges a capacitor and produces very short high voltage charging pulses that break the formed sulfur crystals.
Yes, this will deplete the battery if it's not charged otherwise.