Aside from the professional techs on here, I probably have the most experience trying to trouble-shoot cooling problems within the past ~12 months :redface:
For reference, mine was scenario 2, and was a head gasket problem. Painfully documented in this thread:
http://nsxprime.com/forum/showthread.php?t=141467
Scenario 1: Temperature slowly creeps up over a few laps, and then slowly (quite a few seconds) cools back down if you go up a gear and cruise (at low RPM for a long straight for example). The key here is the slow/steady/predictable change.
Probable causes:
-Blocked / clogged radiator
-bad thermostat
-obstruction somewhere else in the system due to scaling or debris
-missing radiator shroud
-something else about your aero mods is reducing radiator efficiency
Scenario 2: Temperature suddenly spikes (just a short few seconds from normal to red-zone), lifting off, putting the clutch in, etc. may or may not cause an immediate (again, just a couple of seconds) return of the guage to the normal range.
Probably causes:
-Air in the system
-Bad gauge/sender
Air in the system causes the behavior, but it's not the problem itself, but usually a symptom of another problem.
-If you're lucky, it just needs to be bled. You've replaced a hose, or something that has introduced a bubble, you just need to get it out. Follow the book or borrow a venturi bleeder.
Other problems can include (in order of easiest to hardest to fix):
-Bad pressure cap
(replace the cap with one from a parts store, repeat test)
-Cracked coolant tank
(inspect tank, you'll see residue, or you'll see coolant under the middle of the car just in front of the rear wheels)
-Heater Core failure
(You'll smell coolant in the car, the inside of the windshield will be hazy almost all of the time, and in bad cases the passenger floor mat may be damp)
-Head Gasket
(The car may or may not white smoke out the tailpipe. It may also push coolant out the overflow tube if it combustion gas is getting into the coolant system...check the coolant level or install a catch bottle to confirm)