I believe to "live a year in a minute" means that at a time of serious jeopardy or in a life threatening situation, you live all the best or most precious moments of your life in an instant. It is in that moment, or instant, that you think of everyone you love or have ever loved and question whether or not you will ever see them again. Although I have never been in that situation, but in "The Most Dangerous Game" it is a terrifying thought to be in the position of Rainsford and hunted like an animal.
Think of it in this way: someone is involved in a serious accident and the story he tells after it is clear he will survive it is that his entire life passed before his eyes and he witnessed all those years, those events, those people within seconds. It is the same with Rainsford. He experiences much during the time he is being hunted by Zaroff. Some of that is sentimental--family he'll miss, things he'll not do again--but much of it is every bit of survival knowledge he's ever picked up from books, friends, Discovery channel, or seasons of SURVIVOR. These things not only help him to physically deal with his situation, but they also help him stay focused and sane. Going insane at this moment and panicking would force him to make a fatal error...one he knows he can not afford to make.