Born in Wakelon, North Carolina in Wake County, 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 defeated 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.