1. 'President' is a role/rank, and we often don't need an article. Here are some more examples 'He was promoted to sergeant' or 'She works as assistant manager in this department'. 2. 'Action' is an uncountable noun here, so it doesn't have an article. 'What we need is action, not debate'. Also, 'person/man of action' is a set phrase.

Difficult question, I can see why you are confused. 1) I think that with being and becoming titles and ranks you don't need an article: To be come president (of the United States), to become King (of Sweden), to be promoted to general etc. Probably best to learn this as a set phrase. In general, if you use "the" it refers back to something you were just talking about. For example you could say, "He was a member of the Council, and then he became the President" - here you would mean he is THE president of that specific council you were just talking about. 2) Action is a general concept here so it does not need an article to describe it, as it describes the man's inherent characteristics: a man of integrity, a woman of importance, a enemy of poverty. If you said, "a man of the action", then a native speaker would immediately ask you "What action are you talking about? What specific action did you just mention?" Plus no one would say "a man of the action" anyway. In English, general concepts don't tend to have an article. For example: "Honesty is important", never "The honesty is important". Also, in your third sentence, you should have written: "I am interested in THE words "president" and "action" ", because here you are referring back to two specific words that you just mentioned.

First: Why can these words be used without an article?.... I am interested in the words... Second: The so-called zero article can be used with names and general descriptions (and more). Here 'president' means the general description of the position anywhere. The moment you define it, it would become 'Would you want to be the President of the US...?' A 'person of action' is again a general description for someone who doesn't sit around and waits for something to happen but takes things in his own hands. It doesn't refer to a particular situation.

If I would REALLY need to become a president I could achieve it ))