No, not directly. The Xbox 360 uses a completely different architecture (PowerPC vs. x86) and disc format. Original Xbox discs cannot be read by a 360 drive.
Microsoft released software emulation profiles for ~460 original Xbox titles on 360. This required:
This service was discontinued in 2007. No new titles were added after November 2007. If you didn't download profiles then, you cannot add them now (Xbox Live for 360 original titles is deprecated).