The short answer is , there is no official release of Cadillacs and Dinosaurs for the Xbox 360, nor does it work natively on the console. Why It Doesn't Work Capcom Licensing Issues
The Xbox 360 Era: During the Xbox 360's lifespan, Capcom released the Capcom Digital Collection and Capcom Beat 'Em Up Bundle, but neither included Cadillacs and Dinosaurs.
dino.zip — CPS1 version)The game remains a victim of ’90s licensing deals that no lawyer wants to untangle. Unless General Motors and Mark Schultz agree to a resurrection (like Disney did for The Simpsons Arcade), Cadillacs and Dinosaurs will stay trapped in arcade history.
Rare and expensive original arcade cabinets still exist for collectors.