PIII 933 and GF3 Ti200 are a nice match, a slighty faster CPU wouldn't hurt, maybe a Tualatin, if your motherboard has support for it. Or change the motherboard and the CPU, a good socket A board and any Athlon XP should be faster, easy to find and cheap.
Try Quake 3 on your machine, Unreal Tournament works better with 3dfx cards (glide). If you keep the GF3 try OpenGL and D3D, see what's better, OpenGL should be the best, but it depends. Use the latest patch for UT, I believe it's 436.
If you play a lot UT, then try you need a 3dfx card, V5 5500 would be nice, but it's getting rare and very expensive. V3 3000 is relatively common and you might find one cheap (if the seller is not aware that's a 3dfx card). If you can't find a 3dfx card, then a faster Nvidia/ATI card might help a lot in getting better framerate, but a faster CPU would also help.
If all fails, remember that UT runs without problems on modern hardware and operating systems, try it on you main machine. I have it installed on my main machine and it runs great in 3840x2160, all settings to max, also with high resolution pack installed.