Born at Fort Bragg, near Fayetteville, North Carolina, where his father was stationed in the military, Daub grew up in North Omaha. He graduated from Benson High School before receiving his B.S. from Washington University, St. Louis, Missouri in 1963, and his J.D. from the University of Nebraska–Lincoln College of Law, Lincoln, Nebraska in 1966. He served in the United States Army as an infantry captain from 1966 to 1968 during the Vietnam War.
Daub settled in Omaha, where he went into private practice of law. He also became active in the Republican Party. He ran for the United States House of Representatives in the Omaha-based 2nd congressional district in 1978, losing to incumbent Democrat John J. Cavanaugh III. Cavanaugh declined to run for re-election in 1980 and Daub ran for the seat again, defeating Democrat Richard Fellman. Daub was reelected three times, in 1982, 1984, and 1986.
In March 1987, Nebraska's senior U.S. Senator Edward Zorinsky, a Democrat, died of a heart attack and Governor Kay A. Orr appointed businessman David Karnes to fill the seat. Daub challenged Karnes in the Republican primary for election to a full term in the Senate in 1988, but lost by nine points to Karnes, who then lost the general election to former Democratic Governor Bob Kerrey. In 1990, Daub challenged the state's other Democratic senator, J. James Exon, for reelection, easily winning the Republican nomination, but losing the general election by a substantial margin to Exon.
In 1995, Daub won a special election for Mayor of Omaha following the resignation of Mayor P.J. Morgan, and was narrowly elected to a full term in 1997, both times defeating city councilwoman Brenda Council. In 2001, he was defeated for reelection by insurance executive Mike Fahey in a close race. Daub then served as a member of the Social Security Advisory Board from 2002 to 2006 and joined Missouri-based law firm Blackwell Sanders Peper Martin LLP (now Husch Blackwell) in 2005.
In 2007, Daub briefly ran for the U.S. Senate seat being vacated by Senator Chuck Hagel; he exited the race to offer an early endorsement to Mike Johanns, former Nebraska governor and USDA secretary, who won the seat.
On November 10, 2008, Daub announced he was running again for Mayor of Omaha. On April 7, 2009, Daub won the most votes in the mayoral primary. But on May 12, 2009, he lost to Democrat Jim Suttle in the race for mayor, with unofficial final results of 48.7% to 50.7%. In that election, Democrats also gained control of the city council.
Daub served for five years on the board of Omaha's Metropolitan Entertainment and Convention Authority (MECA). On November 6, 2012, he was elected to the board of regents of the Nebraska University system.