How do aircraft batteries power the engine/APU?
Yes battery power will supply both the APU's igniter system and its starter/generator (a single motor that switches to generator mode after start, similar to turboprops).  So you need the APU to start the engines (notwithstanding other bleed sources) but you need the battery to start the APU (or an external DC power source - the APU will be fed from a DC bus; whether the bus is being energized by a battery or a DC ground cart, the APU doesn't know or care).