The 22-year-old won his first medal in the 10m platform event, finishing with a score of 543.80.
China's Olympic champion Aisen Chen took gold with 580.60 while silver went to compatriot Hao Yang (574.80).
Daley then teamed up with 20-year-old Grace Reid to finish third in the mixed 3m synchro behind pairs from China and Canada.
Their best dive came in the third round when they scored 69.30 points for their inward two-and-a-half somersaults piked.
Britain's Tom Daley won two bronzes on the final day of the Diving World Cup event in Kazan, Russia.