Any renewable or nuclear energy source would make it carbon neutral. It doesn't have to be free, the fuel doesn't cost that much in the first place, so a modest percentage cost increase is fine.
We know that making the methane isn't super-expensive, because we've already used the Sabatier process commercially to make methane from CO2. That used carbon-based fuels to get the CO2; to really go carbon-neutral it'd have to be biofuel, or we'd need to pull CO2 from the air using direct-air-capture machines.