My opponent will be out of town and away from his computer for a week. How can I pause the game so as not to run out of time?

The short answer is, you can't, but you don't need to.
The game has 3/7 time controls, which means he has 3 days to make each move, and he has a reserve store (timebank) of 7 days. At this point he has just over 1 day left of his move time, so the skull won't appear for another 8 days. Even when it does appear you do not have to click on it. This site doesn't have automatic termination for standard games. Once he comes back, even if he's used up all his reserve time he will still have 3 days for each move.