280ZX and 280Z ECUs different. Also, early and late 280ZX ECUs different and have altered ancilliary parts to suit - ballast resistor to the injectors is a different design for example. Also ECU is matched to auto or manual car.
As noted, fuel pump operation is externally controlled by relays and should prime the system then stop, restarting the pump once the engine fires / has oil pressure - separate terminal on the 280ZX oil pressure sender.
If you have a part number on your ECU, I should be able to cross reference it for you.
One further thought - there may be a flying lead on the ECU to receive a "run" signaal from the coil / distributor. Is that connected? Nothing works without it....
Is the ECU power relay working OK? i.e. is the ECU getting a correct set of power signals? This relay is a common failure on the 280ZX. Also is voltage strong? Low battery voltage will cause the ECU to misbehave.