Yes, a 48V lithium battery on board. It gets charged by the generator/starter when decelerating, like a hybrid vehicle. It can also add power for acceleration.
As for how long it can run the compressor, I guess it really depends on many factors. It is a quite complex system. But if the 48V battery SOC starts getting low it will start the gas engine.
Even the cooling system is better designed for this as there is no longer a condenser on our cars. There is a high temp and low temp cooling system, each with their own rad and coolant. The A/C refrigerant is cooled in a refrigerant to water cooler (looks like a transmission oil to water cooler) on the low temp circuit, and that coolant is circulated by an electric pump.