Is it true that the Prophet (peace be upon him) did not disapprove of Aisha's (may Allah be pleased with her) statement: "Whoever angers you shall enter Hellfire," and how can one deal with this if every sin angers him?

A'ishah's (may Allah be pleased with her) statement: "Who angered you, O Messenger of Allah? May Allah admit him to the Fire," could be a supplication or a statement of fact.

If it was a supplication, then it appears to be a prayer from A'ishah against whoever angered the Prophet (peace be upon him), considering that whoever does so might be a hypocrite.

If it was a statement of fact, and the Prophet's (peace be upon him) silence was an affirmation, then this hadith is considered one of the warnings, and the one who commits such an act is subject to Allah's will. If He wills, He may forgive him, and if He wills, He may punish him. This is similar to all other sins that are not disbelief, as a monotheist will not abide eternally in the Fire; even if punished, he will be brought out of it into Paradise.