As long as at least one permanent you control is the specified color, the ability will "work" and grant this creature the bonus. Otherwise, it won't have the bonus.

Whether Cliffrunner Behemoth has lifelink matters only when it deals damage (that is, at the point when its combat damage resolves, or at the point when a spell or ability that causes it to deal damage resolves). For example, if your only white permanent is a creature that receives lethal damage at the same time Cliffrunner Behemoth deals damage, the lifelink ability will still apply and you'll gain life.

Whether Cliffrunner Behemoth has haste matters only when attackers are declared. Once it's been declared as an attacker, it doesn't matter if it loses haste.