Rust's mother died when he was five, and his father was an officer in the United States Navy. Therefore, Rust lived with an aunt in New York City, where he studied at the Neighborhood Playhouse at 340 East 54th Street in Manhattan.
In 1957, Rust was chosen as a replacement in Eugene O'Neill's Long Day's Journey into Night, with Fredric March and Florence Eldridge, March's wife. The play was performed in the prestigious Helen Hayes Theatre. In 1958, Rust signed a contract with Columbia Pictures, along with other young Hollywood aspirants Michael Callan and Yvonne Craig. He appeared in 1959 in The Legend of Tom Dooley, a Western film starring Michael Landon, unrelated to the popular song "Tom Dooley" released the previous year by The Kingston Trio. In 1959, he was cast with Roscoe Ates and Andy Clyde in the episode "Well of Gold" of the NBC children's western series, Buckskin, starring Tom Nolan. Rust was cast in 1960 in the crime drama film, This Rebel Breed, about rival narcotics gangs in high schools.
He starred in 1960 as Dobie with Randolph Scott in the Western film Comanche Station. His films in 1961 included william Castle's Homicidal, a thriller starring Glenn Corbett, and Underworld U.S.A., a study of revenge starring Cliff Robertson. He starred as Oliver in the 1962 film Walk on the Wild Side with Barbara Stanwyck and Laurence Harvey. During this period, Rust also appeared in several western television programs, including Black Saddle with Peter Breck, The Man from Blackhawk starring Robert Rockwell, The Rifleman starring Chuck Connors, Gunsmoke with James Arness, Tales of Wells Fargo starring Dale Robertson, Johnny Ringo with Don Durant, and Have Gun – Will Travel with Richard Boone.
In 1962, Rust procured the role of the 24-year-old attorney Hank Tabor on Sam Benedict, with Edmond O'Brien. More than sixty competitors sought the role but Rust prevailed because of his acting experience and his educational background. The series, which ran for twenty-eight episodes, was slated against the new variety programs launched by Roy Rogers and Dale Evans on ABC, which proved unsuccessful too, and Jackie Gleason on CBS.
Rust appeared as James Vining in 1975 on the ABC Daytime soap opera General Hospital.
After Sam Benedict, Rust appeared in various television programs, including Gary Lockwood's The Lieutenant, which occupied the former time slot on Saturday nights held the previous year by Sam Benedict. He guest starred on three episodes of Have Gun – Will Travel starring Richard Boone, two episodes of Perry Mason with Raymond Burr, Bonanza, Christopher George's The Rat Patrol, and Cade's County with Glenn Ford. In 1969, he appeared in a Roger Corman biker cult film entitled Naked Angels. His last named role was in 1990 as Sheriff Blanchfield in the film Double Revenge about a bystander who goes on a rampage against a bank robber.
