NOT a bad pump.
There are TWO ways the fuel pump is turned on.
1. The ECM via the relay. The ECM will power the relay for the prime, and also anytime the engine is turning over.
2. The Oil Pressure Gauge. This also can supply voltage to the fuel pump. It is on whenever the engine has oil pressure.
These two ways are redundant to each other. So if one fails, the car still runs on the other. When the ECM/relay side fails, you lose the "prime" shot, so you have to crank the car until the oil pressure builds, and then it fires right off. You might have noticed it takes longer to build the oil pressure when the engine is warm vs cold.
When your worn engine is hot it is pretty much impossible to build enough oil pressure just with cranking I am thinking. Start it on starting fluid, and you then have enough oil pressure to turn on the switch. If by then the engine has cooled enough, it will even idle since the oil is thicker and not making the switch turn off. That is until it gets warm enough to do it again at idle.
Usually when you lose the ECM/Relay link it is because the relay socket gets corroded enough so as not to make good contact with the relay. That is where I would start. If you can read schematics, and use a multimeter, you can see if you are getting the +12v from the ECM to the relay socket. Do you hear the relay clicking? Yes the ECM could be bad also and not supplying the +12 to fire the relay. Also a possibility is the wiring.
One thing you can do as a 'redneck' fix is to send power to the pump via a wire from the ALDL fuel pump pin G connected to +12v
This will make your fuel pump run whenever you supply it voltage

Note - my picture and Hudini's are the same, just upsidedown from each other. I do like his better btw
You have pretty low oil pressure I assume, so much so that sometimes during hot idling that the gauge goes down to the red, correct?
So - clean - and/or replace the socket for the relay. That's my guess.
[This message has been edited by phonedawgz (edited 05-30-2010).]