One should have confidence in himself, otherwise, he will boost the confidence of others but not himself.
This is correct. "Himself" refers to a person. He will not boost the confidence of himself.
One should have confidence in himself, otherwise, he will boost the confidence of others but not his.
This is correct because a missing noun is implied. He will not boost his confidence.
If I were to say it, it would be like this.
One should have confidence in himself. Otherwise, he will boost the confidence of others, but not his own.
This is style. Short sentences are better than run on sentences.
"His own" sounds more natural to me than "His". "His own confidence" is implied. I can't really say why it sounds better. But ending with "his" sounds too abrupt.