In your initial post, you stated that the engine cranks, starts up, revs to about 1800 rpm and then dies. This suggests that the engine is getting both fuel and ignition, at least during the initial start sequence.
The NSX service manual provides the following description of the start sequence:
<!--[if !supportLists]-->1. <!--[endif]-->When the ignition switch is initially turned on (II), the ECM supplies ground to thePGM-FI main relay that supplies current to the fuel pump for two seconds to pressurize the fuel system.
<!--[if !supportLists]-->2. <!--[endif]-->Whenthe engine is running, the ECM supplies ground to the PGM-FI main relay that supplies current to the fuel pump.
Since you have upgraded the fuel pump and removed the fuel pump resistor, I am going to hazard a guess that you have removed or bypassed the fuel pump relay and are running the fuel pump directly from the FI main relay – so we can eliminate the fuel pump relay as a potential problem.
Since your engine starts and then dies, I am going to speculate that the fuel pump may be getting the initial 2 second turn on prior to cranking which pressurizes the fuel line, the engine then starts during cranking but the FI main relay is not getting ground through the ECM once the engine starts running and then dies from absence of fuel. The best way to check this would be to measure the pressure in the fuel rail. If the fuel pressure falls immediately after the engine starts you know that the fuel pump is not being energized after start up (or you have aspectacular blockage in the fuel line, which I doubt since you did have a period where the car was working for a while). In the absence of a fuel pressure measurement, you could check the voltage at the fuel pump to see if you get the initial 2 second on when the ignition switch is turned to II, then see if the fuel pump gets power (briefly) after the engine starts and then dies. Measuring the pressure at the rail would definitely be the more definitive test. A pressure measurement would also identify problems with the fuel pressure regulator (or its setting) which could be causing a fuel starvation problem.
If the fuel pump is getting power after the initial start (or you are maintaining fuel pressure), then I am at a loss as to your potential problem. If the fuel pump is not getting power after the initial start it suggests that there is a problem with the ECM or its related systems (might be no tach signal – but I think that would prevent the initial start-up since the ECM needs a tach signal for ignition).
The preceding comments are based upon the operation of the OEM ECM. You stated that you had an AEM ECM; but, I am going to hazard a guess that the AEM will replicate to some extent the start-up sequence in the OEM ECM.