You might be able to do it in four days but that sounds like an awful trip. I think it's better to go down there when you have a little time to enjoy the place. There's a great market in Tlachichuca on sundays.
I climbed Orizaba during Spring Break and I had time to visit Oaxaca for 4 days. I started in Chicago with no acclimation. I flew to DF Friday night and made it to Puebla on the last bus leaving the airport. By early Saturday I was in Tlachichuca and arranging an ice axe rental and a jeep ride to the refugio. I spent Saturday night near the hut, Sunday I walked around a little. On Monday I hiked up maybe a 1/2 mile and dropped off my axe and some other light gear. Tuesday was my summit day, and it took the whole day. If you have good weather and are well acclimated, you could do it in 3 days from Mexico City but it doesn't leave room for error.
Minimum time is whenever you have acclimatized. It takes longer for some than others.
Some have actually climbed Orizaba the day after landing in Mexico, but that's not a good idea. The Mexican Volcanoes were m first experience with (minor) AMS.
It would be better if you spent at least four days at moderate altitudes before climbing Orizaba, and the more days acclimatizing, the better. You could always spend the 4-5 days time climbing La Malinche or Cofre de Peorte and seeing other sites, so no need to rush up to the highest volcanoes right away.
I think there is a direct bus from Mex City to Tlachichuca, I know there was one back. You could arrive and take the bus (or a taxi). Next day motor up to the hut, one day acclimatization. Day three summit (I started up at 2:15am and was back at the hut at 10:15am) and return to Tlachichuca, maybe even catch a bus that day to Mex City. Day 4 back to an afternoon flight out of Mexico City, the bus leaves very early so perhaps you could make an earlier flight home. Scott is of course right, some people acclimatize faster/slower than others, this schedule allows very little "slop" time. Unforseen problems would result in no summit. If you were really lucky, you could illiminate the acclimitization day and do it in three, but this would allow for hardly any sleep. I think 4 days is very doable, three is also possible without killing yourself.