I did my first track day with the Z in October, Road America. No oil cooler and no Limp mode in any of the 5 20 minute sessions I did. Of course I was in the beginners group so I didn't go "all out". I also took it easy on the long main straight, I shifted up to 6th to give the engine some time to cool down and to let all the big boys in their Corvettes and Vipers pass me. But the one time I pushed a bit harder for a full lap, the oil temp did come awefully close to the limp mode temp. So based on my experience: you should be fine for your first track day, just keep an eye on your oil temp and keep your revs down once in a while to let things cool down.
Oh, yeah, brakes: you do have to upgrade your fluid and pads. I can happily recccomend the Carbotech XP10/8 pads I used.
But as I want to try a little harder next year, I have started to order the parts for my DIY oil cooler, lines are in, cooler is next, i should be done by the time the snow melts