Ah, one of my favorite hobbies - spending other people's money
A coworker had a 99 Civic Si, and I've driven it a couple of times. The car was stolen and stripped to a shell a couple of years ago, and unfortunately, that's all too common for that car.
I agree with basically everything that Karl says about the car. It is absolutely necessary to drive the car like you hate it to get any type of speed out of it. It's sprung fairly softly from the factory, so suspension upgrades are necessary in order to keep it from plowing like a tractor. With appropriate suspension\tire upgrades, that car can be a handling star.
My major disappointment with that car was the brakes. The braking feel wasn't very good, and I've seen other reviews of the car that concur. It also just doesn't have much in the way of rotor mass. Good pads and fluid, and possibly even brake ducts will go a long way, but that car will always be under braked with the factory setup.
Though unlikely with a car that has seen life as a street car, some high mileage b16 motors have a tendency to "oval" out the cylinders, resulting in poor compression, excessive oil usage, and poor power. Again, this is unlikely to be an issue with a street car, but Civic Si's that have seen lots of high RPM duty, like a Showroom Stock racer, are prone to this.
Honestly, put a Harddog roll bar (or a Chris Schimmel bar) in your CS Miata, and I believe you already own a better track car than a stock Civic Si. I understand that you may not want to use that car in that capacity, but stock for stock, I'd say it's superior. IIRC, you guys also have an NA Miata (unless it's been sold). I'd say the same about that car vs the Si.
In this market, it seems reasonably easy to find someone else's "built" track car. Finding something that already has some suspension work, a roll bar, and possibly seats/harnesses seems like a good option. The downsides are harsness in a daily driver role, and hidden wear/tear problems that the seller may be hiding.
Depending on budget, I might suggest anything from an e30 to an RX-8.