If you have "status" with the cruise line, are booked in a suite or concierge level, and/or specifically let the purser know in advance, usually you can make a run for it as OP describes. However, I have been on cruise ships that have docked late because of weather and also been help up because of customs, so it is taking a risk to cut it that close.

Most cruise lines want to get ride of you as quickly as possible as it is a big job to turn the ship around and many now offer early boarding for their "status" guests. That said, because the big ones carry so many passengers in is subject to lines and delays.