He thought that his hour had come and wished to be allowed to meet death like a man.

In indirect speech, the reported clause's tense is changed according to the tense of the reporting verb. Here, the reporting verb is in past tense so the entire reported clause will be changed to past tense. As the second part is an imperative sentence, "let me" shows permission, so "allowed" is used here. As the reporting verb says "thought", "wished" can be added instead of "asked".