A native of Johnson County, Texas, Culpepper was a linebacker for the Texas Longhorns 1960–1962. He helped his teams to Southwest Conference championships in 1961 and 1962. Culpepper was an All-SWC selection in 1961 and a team co-captain as a senior. He is best remembered for his hit (along with Johnny Treadwell) on Arkansas' Danny Brabham in 1962 that forced a fumble at the goal line and spurred the Longhorns to a 7–3 victory and a SWC title.