link postFrom update!
I got it working without downloading it thru steam. Here's the process:
1) If Steam is installed in C:\, either install the game to C:\Steam\steamapps\common folder or copy your UT3 installation to C:\Steam\steamapps\common\"unreal tournament 3" folder (without the quotes).
2) Right click on "unreal tournament 3" folder, select properties and uncheck the "Read-Only" checkbox and click ok.
3) Start Steam and install game thru it. Steam will download some stuff (around 200-300MB) and link the installation.
After this, you can run the game directly thru Steam and see ur UT3 achievements just like you do for TF2