I have a '84 n/a also and I have the same problem.The car runs 30 sec. and dies,it starts right back up and runs for another 30 sec. Did you find out what the problem is?

 

here are some things may not fix the problem but will prevent others down the road:
Get your battery load tested. Be sure you have clean, tight corrosion free terminals on both ends of your battery cables. Be sure the ground (negative) cable goes to a bolt into the frame before going to the starter. grounding through the starter is not a reliable connection. Z's don't like low voltage. causes the electronics to act funny. it is possible to have enough amps to crank but not enough voltage to run the electronics. If you have one size fit all cheapo clamp on terminals they are a problem waiting to happen.

Clean the connectors for the maf or afm, ecu and tps. Deoxit by CAIG is probably the best connector cleaner on the market. spray with CorrosionX after cleaning and before putting together this will help prevent any further corrosion.

put a pressure gage on the fuel ring and see if pressure remains steady as it dies and then what happens. if it is pump quitting then try a hot wire direct and see if that works. after that study the schematics (download fsm from xenon). Might be a faulty relay.

 

In the Engine Fuel and Emissions Control section page EF&EC-27 begins the diagnostic info for your issue, most likely a bad relay, the 30 seconds is pre programmed starting cycle, after that it depends on the relay to continue operation. Older Z's , like 280ZX used 3 different relays to control the fuel pump, from what I can see the z31 uses just one, so it must have a bad side, or bad connection, pull it out and re seat it, just might fix your problem. If not replace the relay (as per Rogerz).