I drove straight up A1A from VB to the space center on Jan 6th. Took exactly 1 hour and 7 minutes. Kept track because I knew someone here on the DIS would ask that question someday.
A1A (not Rt 1) in my opinion is the way to go... A much nicer and more interesting drive up. If you like wildlife you can also visit the Merritt Island National Wildlife Refuge, which is a part of the KSC property (on the south side).