Breathing in cold air can irritate the lungs but the cold air will not permanently damage or harm your lungs or health in any way.
Breathing in too much cold air causes irritation to the lungs and makes it hard to breathe since your lungs are warm and moist and when the cold dry air suddenly hits the warm lungs it causes your lungs to hurt.
There's no danger of breathing in cold air though since our bodies are setup to easily accept this cold air and then warm it to make sure our lungs get the needed warm air.
So while the cold air may hurt your lungs and make it harder to breath there's no danger in it.
But you can also wear a scarf over your mouth to help keep the air that you're breathing in warm so it doesn't affect your lungs.