"Out of the blue" means the same thing as "suddenly." Stick shift is a type of car, where you use a stick to manually drive the car, along with automatic car, where you just switch from park to neutral to drive, etc. Out of the blue = 意想不到 Stick shift = [操縱變速桿（換擋的方法）] 手動換檔 (Stick shift car = 手動換檔車)

The phrase is "out of the blue". You do need "the" in the phrase. It means "unexpectedly", and refers to a lightning bolt coming from a clear blue sky. "Drive a stick" means drive a car with manual gear operation. The stick is the gearstick that you use to shift between gears. You can say "he can't drive a manual" (ie. a manual transmission car).

"Out of the blue" has to do with something unexpected. For example, "My friend called me out of the blue." means he called me without my expecting him to. "Stick" as in "driving a stick" means a standard transmission car as opposed to an automatic.