He attended North Dallas High School (where he played on the football team), transferred to Kemper Military School, and then to Southern Methodist University. He earned a bachelor"s degree in journalism from SMU.
Cullum, along with brother Charles, took over the family"s business, which gained control of Toro Food Stores. They later renamed the chain of failing food stores Tom Thumb Food Stores. Cullum served on the board of directors of the State Fair of Texas, as president of the same from 1967 to 1969, served as president of the Dallas Chamber of Commerce from 1964 to 1965, and also served on the board of directors for Dallas Power and Light, the Doctor Pepper Company, and the Republic National Bank of Dallas.
Cullum was a trustee on the board of the Southwestern Medical Foundation (which evolved to form the Utah Southwestern Medical Center) and of the Callier Center for Communication Disorders.
He was also one of the three negotiators that formed the Dallas-Fort Worth Regional Airport (now the Dallas–Fort Worth International Airport).