To claim a draw by repetition (or by the 50-move rule), you press the draw button at a time that the proper condition is met - i.e., when the current position is a threefold repetition or is after the 50th move without capture or pawn move.
This does mean that if it is your move that creates the condition, you must hit the draw button before your opponent moves (unless your opponent's move does not break the condition).
If your clock runs out, you either lose or draw by the rules governing insufficient material; the existence of a threefold or 50-move rule position does not grant you a draw on timeout.
Neither the threefold repetition nor the 50-move draws are automatic; they must be claimed.
How do I claim a draw by repetition?
Also, does it matter who makes the final move as in my opponent or me? If it is my opponent then it will be my turn and I can take my time. But if I make the final move do I have to claim the draw before he moves? What happens if my clock runs out?
Does this also apply to the fifty move rule or is that automatic?