Hello
What is second push? I am assuming ignition, on my 370z clutch in one press turns it on. Second would turn it off. No clutch two pushes only takes it to accessory on.
Is the car trying to turn over, does it ever turn over or just starter noise?
Cars only need fuel spark and air to run, so elimating what it does have is a good start for troubleshooting the problem.
Being that its a new motor, I would be curious if timing was set to TDC, if it turns over then shuts off. I have not searched to see if the z if an interfereance motor, but if it is you can damage stuff if timing is wrong.
If it comes on with a SES light, the immobalizer could be the issue. Remove the batter term for a min and try again. If not I assume a dealer visit to remove error. In the world of Hondas, the k swap requires Imm. to be disabled before they will run, if stock ect is used.
Found this basic t/s link on this forum also::
Why a car won't start - troubleshooting tips