It will not matter as your problem is high intensity efforts. Ride the climb that is causing you problems once or twice a week. If you ride with a HR monitor, do the climbs at 90+% of max-HR, aim for 5 minutes and slowly extend the time to 10 minutes.Specifically in regards to cardio and building lung endurance. (my lungs usually quit before my legs do). Is it better to do shorter rides every day, or a longer ride every other day with a rest day in between?
And for a longer ride, try keep the HR between 60-75% for 2 hours, but not above, just in that range. After 2 hours ride how you want.
Those two elements will develop your aerobic system (Heart, lungs, circulation and muscle). If you want to develop speed and strength, do 30 second sprints on short rises (ca 5%). Just a few, and after some weeks you will be faster. Like magic.