Is “your time is done” often used?
Solution 1:
"Your time is up." would be the much more common idiom, meaning "you've run out of time".
Possibly the author chose not to use this phrasing because "up" is used otherwise baseball, namely "up to bat".
Solution 2:
To answer your second question first, Oishi-san, yes, the repetition of he appears to be a typo.
And "your time is done" does means "your time is finished" or "your time is up"; all of them mean something has come to an end: your career, your marriage, your life, whatever is the subject at hand.