Dan, I agree with Barry. One of the single most important characteristics for anyone to enjoy alpine climbing success is the ability to endure inordinate amounts of pain and discomfort. Certainly a high standard of fitness is necessary, but just as important is a strong force of will. All that aside, here's some more food for thought. You're born with a ratio of fast twitch and slow twitch muscle fibers (power and endurance) that will never change - one does not convert to the other. How you train them will determine which way your fitness level leans: the power side or the endurance side. Some people are genetically predisposed towards bigger more powerful muscles, while others are the opposite, and most of us fall somewhere in between. The ones who are on both sides of the pendulum are in the Olympics sprinting and running the marathon. When training to be in the mountains, identify what you enjoy the most and train accordingly. An ascent of Rainier will require a completely different regimen than if you were going ice climbing around Banff. One thing you can't neglect in either situation is your lactate threshold. This allows your body to exert a higher workload for a longer amount of time without your legs or your forearms flaming out. There are plenty of good articles out there on interval training which you can find that addresses this. Also, when training Courteney is right about doing the hard, powerful stuff first, but think about switching it up every now and again to incorporate the mental game into your workouts. If you're at a climbing gym, do a bunch of laps first to tire yourself out, then hit the hard boulder problems. The crux of climb may not be right off the ground, and if you're mentally prepared to pull a roof while you're at the end of the pitch, cold, tired and hungry, you have a much greater chance of success.
Eddie