Some aircraft like the RJ100 rely on the apu for cabin pressure and airconditioning rather that taking bleed air from the engines when extra puff is needed to get out of places like Innsbruck or Salzburg.
Not sure about the RJ, but the 146 can not take-off at all if using engine air for the packs. So if the APU is u/s (or not used, as our SOPs recommend), the cabin is not pressurised until the engine air and packs are switched on (only done after climb power selected). In this case, the packs and bleeds are switched off by 500' agl on landing. For both T/O and LDG, one engine air is left switched on, but only to drive the discharge valves.
Unpressurised operation, whether deliberate or not, sounds crazy. And it is, but that's the 146!
For non-APU starts, we use the GPU for all 4 engines, or else start one on GPU, then cross start the rest (usually use the GPU for all 4 unless in a hurry and want to start the rest during the push). The 146 doesn't use air start, each engine is fitted with an electric starter motor which turns the N2 shaft. It makes no difference timewise, whether we have an APU or not.