Generally speaking, you'll be in much better shape than the average person, but mountain climbing requires also a good amount of upper body strength, balance, coordination and experience.
I was recently chatting with a ranger at a rescue station in the Grand Canyon and he told me that the typical profile of the person that gets in trouble hiking the canyon is middle-aged, male, marathoner. He explained that marathoners think that 20 miles are a joke and underestimate the difficulty of hiking on uneven terrain, on exposed paths, in the heat, with little water.