Nietzsche said that you can't dodge your challenges and
hardships in life. Things like alcohol is bad and makes you
weaker. He only drank water and occasionally milk. He was saying
that you can't say "everything is going to be alright."
Don't lie to yourself and you will succeed in life.
Nietzsche lived a life of chronic pain and agony. He would suffer migraines for days, vomit and have allergy problems which inspired him to move to Zurich. HIs ideas around pain suggest one needs to embrace pain in order to overcome it and feel alive. There is tremendous truth in struggling through obstacles to feel victorious when you've succeeded but that's not the whole story.
Often, pain should not be thought of as something one should conquer but rather something that must take its course. For example, bereavement. Losing a spouse or child is one of the worth pains imaginable but having one's mind on overcoming the pain will distract from coping with the loss which leads to recovery.
So no, not all pain should be thought of as something one must conquer in order to be stronger in the end. Often times, that strategy results in a weaker, more vulnerable personality since a strong person can endure pain not just overcome it.