You can lessen the hesitation by getting proper bolt ons, but really some sort of EMS will get rid of it the easiest. The problem is an overly rich air/fuel ratio. Proper bolt ones and added boost will help hide it, but an EMS system will fix it. If you can, save up and get a programmable EMS system if you want to tune yourself, or a Unichip if you want plug and play.
If you want to go the Bolt on Route get this:
First $500:
Exhaust $100-900 depending if you buy new/used/part/full system - Midpipe that replaces the catalitic converter. I bought a used Apex 2.5" catless down/midpipe one for $150 and love it.
New Vac lines ($10) - replace all your old vac lines as the stock lines in an MSP get very brittle and crack.
Front Motor Mount ($90) - Awr sells an upgrade for your front motor mount which help eliminate wheel hop ad transfer more power to the ground. The higher the dourometer, the more it will cause you car to vibrate. Get the 70 durometer ones if you don't want increased vibrations
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Next $500:
New fluids and tune up for your ride. Unless you know the previous owner it would be a good idea to replace all the fluids. Get some redline transmission fluid and at least check the condition of all the others
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Next $500
Upgraded intercooler. If you plan to run 10+ psi or just really like the look, get a FMIC. They run $900-1200 new and will make a big difference in your hesitation issues, especially in hot weather. You can find a used one for sale, or just get a core and build your own pipes for under your $500 limit.
If you want to appear more stock looking, or don't plan on raising the boost, get a set of Hardpipes and an upgraded SMIC. YOu can probably find this combo for sale here too, not sure on total costs.
Manual Boost Controller ($50)
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At this point combined with your CAI you should be good to run 8-10 psi with little to no problems. I've been running 8-11 psi for about a year with the above modifications. The only thing that's gone wrong with my car is I think I have broken the stock rear motor mount with that extra power.
This will definatly make it faster than a stock S2000.
Then the rest is up to you. You can either spend your next $500 on a better boost controller that you can adjust boost by rpm or gear, or save it for an EMS system. Personally I'd get the EMS system. I've ridden in an MSP with identicle mods to mine but he had the Unichip. There was a night and day differance in power (of course he's running 2lbs more boost too) but his power was constant throughout the powerband while mine tends to hit a flat spot between 5300-6000 rpms. I'm saving for it now.