I have an 83 280zx, (130,xxx miles) withprecisely the same problem.
It starts up,.. barely I might add, and idles roughly with little power to accelerate, spitting and backfiring as I try. Once I manage to get it over about 3500 RPM's it takes off like it should, speeding down the highway, all the way to red-line if I want. After about 45 minutes, however, it slowly chokes itself and can't be started again until the engine cools down, (it isn't overheating, it just dies after reaching operating temperature). According to my Chilton's and Hayne's manuals, these symptoms describe a vapor-lock situation, but doesn't explain the same low-pitched bogged-down hum of the fuel pump,.... leading me to suspect that the engine warming up isn't the problem. I actually propose that maybe the fuel-pump, after warming-up, begins to give out.
I don't know about you, but I hate fishing for the problem by unneccesarily replacing part after part.
So, though I haven't quite figured out what the problem is myself, I'm certain in my case that it wasn't the fuel injectors because I recently replaced all of them. One way to check to see if your injectors aren't seized is to listen to them. Take a long screwdriver or something comparable and place the tip firmly on the top of an injector. Now press your ear to the handle,... you should hear the injector crisply snap as it fires. Try this on all of your injectors. When you are satisfied that they all are firing, begin disconnecting the wires, one injector at a time and listen to make sure the engine RPM's slow down and idles rougher. Replace the wire and the engine should return to a smooth idle. If removing the wiring from a particular injector causes little or no effect, then you probably have a bad injector,... make sure of course that you don't just have a fouled or bad spark-plug.
Another common cause of backfiring is a bad oxygen sensor, located on the exhaust header. You are supposed to replace these every 30,000 miles or so. Not a very expensive item, about $30.
Check your PCV valve and hose located under the intake manifold. You may not have to remove this, just pull the hose, (engine cold) and use something not metallic like a toothpick to push up on the ball inside. It should move up freely and then fall back down on its own. I've never replaced mine, just took it off and cleaned it with carb-cleaner once.
Your EGR valve is located near the rear of the engine. Make sure the gasket isn't leaking where it bolts to the manifold. With the engine running, (in Park) push on the accelerator while reaching under the valve. You should feel the valve open and close when you rev the engine.
I'm about to replace the pump myself, lucky I know of a wrecked ZX in the local salvage yard, (I'm cheap)! If this dramatically fixes my problem, I'll let you know.
Otherwise, good luck pal!