Crudup was born in Wakelon, North Carolina in Wake County, the son of Elizabeth (Battle) and Josiah Crudup, a Baptist minister. Crudup attended a private school in Louisburg, North Carolina and then Columbian College (now George Washington University) in Washington, DC. He studied theology and was ordained as a Baptist minister, which was his profession his entire life.
Also a farmer, Crudup was elected to the North Carolina Senate from Wake County in 1820, but was forced to vacate his office because the state constitution at the time forbade "a minister of the Gospel, while exercising his ministerial functions, to hold a public office." 
In 1821, he was elected to the 17th United States Congress and served for one term, from March 4, 1821 to March 3, 1823. Although he ran for re-election in 1822, he was narrowly defeated by Willie P. Mangum and returned to farming and the ministry. Crudup was a delegate from Granville County to the 1835 North Carolina Constitutional Convention, and died in Kittrell, North Carolina in 1872; he is buried in his family cemetery near Kittrell. Among his descendants is actor Billy Crudup.
- NC Manual
- "Daily Dispatch: Group works to save historic Crudup home site in Kittrell". Archived from the original on July 18, 2017. Retrieved July 18, 2017.
- United States Congress. "Josiah Crudup (id: C000953)". Biographical Directory of the United States Congress.
|U.S. House of Representatives|
James S. Smith
| Member of the U.S. House of Representatives
from North Carolina's 8th congressional district
Willie P. Mangum