Sands Of Time does work with x360 controller but there are two problems. You still have to use mouse and keybord to use the menu stuff(starting new game, loading, saving, options, pausing the game etc.) and there are no x360 button prompts. First thing is setting up the controller in options, you can use whatever settings that suits you but for good start i reccomend just original ps2 layout, you can find it here:
Sands of Time for some reaseon allways will display the keybord prompts even if you are using controller, the only way around it is to bind controller buttons to keybord section(you have to use controller buttons as your primary input)
After that just download this file and replace the old one with this one in the main installation folder. After that all buttons prompts will be compatible with x360.(to be honest almost all becouse text in game will always reffer to cammera as mause, but all the rest works as intended;)

Forum is forbidding me to post any links but i did put the same exact message on steam forum so you can get the file there;)