So I have a question about boromir. Do you raise his threat only if you are readying for a second use? Or any time you ready him, such as in the refresh phase at the end of turn.

Only when you use his Action (readying him for a second, or third, or fourth use..). Actions are voluntary effects that you choose to trigger. If you don't have a choice, the text will say "Forced" or, in the case of a passive effect, it won't have any label like "Forced, Action," or "Response." See the spirit Glorfindel for an example of a card where you don't have a choice.