NSXFTW,
I myself have the same issue. I do believe it is a problem and not a normal variant. Yesterday, I met up with a few other NSX owners and their cars started just like any of my other Hondas...quickly. Basically, their cars cranked for a very short duration (almost <1 sec) and started up.
I've spent a fair amount of time searching the NSX prime forums, however, have not found a satisfactory solution.
For me, I always start my cars by allowing time for the fuel pump to pressurize. Then I crank the motor. Always, the first time I start the '91 NSX (unmodified drivetrain) for the day, it has a lengthy crank time. Literally, almost five times as long as any of my other Hondas.
This was an issue for the prior owner and he took the car into the dealership to do a compression or leak down test (can't remember which one) and the results were quite good (all cylinders were nearly identical and well within normal limits). Again, can't recall the values nor do I have quick access to the results right now.
Here is what I've done and there has been no remedy:
-I, too, replaced the battery which is now about 3weeks old.
-I've removed the main relay and resoldered all the joints.
-Cleaned the throttle body which was quite dirty. The car idles even smoother now, but doesn't start any better.
-My ignition switch seems to be working fine and the car cranks well at a normal speed so I can't imagine this being the issue.
-Battery cables on tight, but have ordered new ground terminal in case there might be some anomaly that I simply can't visualize.
After all this, the car starts the same. Now here is one interesting behaviour that I haven't seen posted before and maybe one of the experts might offer some insight. If I shut the car down shortly after warmup and the car never goes above idle, the subsequent crank time is long. However, if I allow the revs to reach 3-4K RPM after warmup, shut the car down, and restart the car, the crank time is like any other Honda, quick (~under 1 second). Now this is not an occassional phenomenon, rather I can reproduce this anytime. Also, the car, otherwise, runs incredibly well.
So I wonder, could this behaviour be due to a bad fuel filter? Fuel delivery? I can't imagine the fuel pump being bad since it seems to run otherwise quite well.