Recently had a very similar issue with a newly purchased and at the time, non running 93 1FZ FE.
Eventually after getting her home we got her to start but run very, very rich. Not really knowing what I was doing, I was tinkering around with the Air Flow Meter, disconnected the plug and then the screws holding the connection housing to the alloy case. Tried removing said housing but couldn't (I now know why).
Put it all back togrther, tried to start and it fired but then died straight away. After about a week of doing all the normal checks, fuses relays igniter etc etc., and still no closer to the no start solution, I came across another used AFM which I wanted to replace anyway because I felt it was the cause of the original issue.
After replacing AFM and nothing else, she fired straight away and now she's running perfectly.