Kirk Douglas did not ride a chariot in a movie. The famous cinematic chariot race is from the 1959 epic film Ben-Hur, in which Charlton Heston starred as Judah Ben-Hur.
Kirk Douglas was actually offered the role of the antagonist Messala in Ben-Hur but turned it down because he wanted the lead role of Judah Ben-Hur. He felt he was passed over for Charlton Heston. The experience motivated Douglas to develop his own Roman-era epic, Spartacus, which was released the following year in 1960.
Spartacus focuses on a slave rebellion and features gladiatorial combat and large-scale battle scenes, but does not contain a chariot race sequence like the one in Ben-Hur. Douglas produced and starred in Spartacus, a highly successful film in its own right.