Suppose I'm buying ads for keyword x and suddenly x becomes the biggest thing in the world. Will my budget run down faster?
Will the price stay at the same rate I purchased that keyword on or will change with the popularity?
As more people bid on an phrase the higher the cost of the phrase will be. So, yes, if that phrase becomes very popular your budget will run down faster as you can expect more bidders to enter that market.
Provided that you are at a sufficiently high Quality Score for that keyword (7+), then yes the budget will whittle down faster.
If the QS is under 7, then you can work on that first. This should reduce your CPCs.
Also, I would suggest that you try Dimensions report in Adwords to see when is the time that you get the most searches and clicks for that keyword and bid aggressively during that time.