I am a big believer in fittings, although I have never been fit myself (only got serious about the game a year ago. Watching TY (TXG mostly), I have seen what it can do for some people. When it comes to shafts, I am less of a believer. Shaft weight and stiffness certainly play a role in getting a great fit, but more important is gapping, lie angle (I lump length in with this because the correlation is heavy), and the proper heads. I am by no means an expert on this stuff, but if I do go for a fit, those 3 things are what I am going to ask about the most.
If you are going to buy new and they offer you a crazy good shaft (usually at a crazy price) based on numbers that are marginally better, I would ask to try the "stock" shaft closest in weight and stiffness to the exotic. They will be different, but not enough to warrant the price in most cases. There are outliers where the exotic is a big upgrade that warrants the price, but those are few and far between.
A lot of practice greens don't have the same conditioning as on-course greens. My suggestion would be (Step one) to try for the practice green, but only for 5-10 long putts (you can play many more short ones if you wish). You don't want to get too grooved in if the on-course greens are substantially different. (Step 2) After that, you pay SUPER close attention to the speed on green 1. Remember that green speed also dictates amount of break. If you get to hole one, and it's slower, and you miss high side, remember that the high-side miss wasn't because you misread it, it is because the green is slower. I have been on fire with the putter lately, and I think that this is why. Just be very perceptive on the first green, basically.
For my pre-round, I am going to vote for the least amount of practice, only because I make lots of tee times straight after work, and I literally don't have any time to practice pre-round. Most of my weekend rounds I get to the course early though. 20-30 minutes is usually the mark for me, forgoing range balls (gotta pay for them thangz), and going straight to the practice green to get my chips going. I usually putt for 3-5 minutes and spend the rest chipping. As I am a player that doesn't hit the green in regulation that often, I want to make sure that when I am close, I have my strokes dialed in for those short yardages. I usually only use the 52 degree, as I can predict what "distances" I will get on the greens for the rest of the round.