The problem is not fitness, it is how quickly your body acclimatizes, a process which takes time yo-yoing up and down the mountain. That said, there are some ways to cut down the time. I ran into a French party who were able to climb Denali in 8 days round trip from the Kahiltna air strip without prior acclimatization by taking Diamox prophylactically.
I climbed Rainier once with a fellow who climbed Aconcagua, airport to airport in 60 hours
. He had previously climbed Everest and Lhotse with only a single bottle of supplemental oxygen. He is sponsored by Hypoxico http://www.hypoxico.com/
, a company that makes tents and masks which simulate high altitude (low air pressure). Brian would exercise with the mask which would pre-acclimate him. <cautionary note>
Be advised I am not advocating either approach. These were all very strong, experienced alpinists who has spent a lot of time at altitude. You can really get yourself into trouble by trying to climb too high, too fast. Experienced climbers have died from HACE on the summits of Rainier and Shasta (14,000 ft) when they got pinned down by weather. If one hasn't done any serious mountaineering before, trying to speed climb a nearly 7,000 meter peak is a ticket to misery, failure, and potential life threatening conditions.
Bear in mind that the 'average' climber who successfully climbs Aconcagua has prior mountaineering experience. There are soft skills, self preservation skills, that are best learned in more forgiving mountains. Every Mount Rainier guide has had clients who were 'fit' marathoners but couldn't make it past Camp Muir. A marathon is a sprint compared to mountaineering. A more reasonable objective for your time frame would be the high volcanoes near Mexico City. Not as high as Aconcagua, but higher than anything in the lower 48. </cautionary note>