Noun. A relatively brief period of time leading up to a particular event. Synonym of lead-up. A Google Ngram shows a spike between 2003 and 2005, supporting my impression that wide use began in reference to the months before U.S. invasion of Iraq, a period in which the imminent military action was obvious to everyone in the country. (Usage also sharply increased in 2007 and 2008, which I attribute to the popularity of not one-off Britishisms.) Also, particularly in journalism, an increase, as in a run-up in gas prices; used as an alternative to hike, spike, and other elegant variations. “To nervous allies, those words echo the run-up to the Iraq invasion, which began three years ago today. But Iran is not Iraq. (New York Times, March 19, 2006)/“The Packers’ report is more than a novelty in the run-up to their playing the Pittsburgh Steelers in the Super Bowl on Feb. 6.” (Richard Sandomir, New York Times, January 27, 2011.)
notoneoffbritishisms.com 
Possibly from foreign news reports? If not broadcast directly in the US then from those monitoring such reports.
I hear this phrase used in the UK, but if you had not suggested otherwise, I would have guessed it was a migrant from the US, not vice versa.
Surely it comes from cricket: a bowler’s run-up to deliver the ball. (A baseball pitcher stands still whilst pitching.)
Agree with Dormouse about the cricket source. Never come across the ‘hike’ meaning in the UK. We do have the verb ‘run up’, used of bills that are allowed to increase before settlement (same, I think, as running a tab) and also when quickly running up a garment on a sewing machine.