If you have multimeter (basic cheap meter all needed) you want test earth & voltage at the 2 wire plug on fuel pump.
ECU controls the switching of the pump on pin 104 of ecu, this switches to ground & is the ground wire on switch side of pump relay, Ecu has initial timed prime when it switches pin 104 to earth & then also switches when it detect engine cranking/running .
If unlucky it ecu control circuit issue when cranking/running > this would be evident by pin 104 not switching ground & no voltage at pump terminal
Other issues could be low voltage due to various wiring faults or relay fault > this could seen by low voltage at pump and if quickly run a temp direct +ve feed to pump fact it cures running issue.
If this checks ok then investigating pump (pump mounting & pickup socks get some right bodges so will be wise check it out) & as ducie suggested adding a fuel pressure gauge would be useful data.
Most of this takes ages to talk about but a minute to do and hopes are you'll stumble on irregularity quite quickly !
doing fault code check & clearing all codes is good idea too . I make sure mine are all checked & cleared as new engine new slate, then can be checked again to see all clear or something amiss & you know results all post build .
* how does it die off, does it smoothly dip down in revs till cut out or a sudden stall or spluttery missfire stall !
Lack of fuel pressure should be more like first scenario .