Wonder how it would fare in the event of a fire. The only escape route would be up, coincidentally the same way all the smoke would go and the fire would spread.
Water leakages would also be a fun little problem. Or heck, even just regular water and sewage systems. In normal skyscrapers, water is liquid when pumped up, and sewage is sent the other way via gravity. Now reverse the order. Supplying the thing with water wouldn't be a problem - you'd actually have to reduce the pressure several times on the way down - but sewage? All that sludge and bits of things floating in the water, and some waste flushed down toilets because people are morons... try to build a pump accomodating for all that, with the same uptime requirements as a regular pump pumping only clean and entirely liquid water. In regular buildings, getting waste out and away is thankfully no problem in case of power failure. Here, everything would seep to the bottom whenever something goes wrong or breaks down. You'd need a small sewage treatment plant to make the sewage sufficiently slushy to be pumped away. The smell would not be very pleasant.
You could also mention insulation. Above ground in Mexico, keeping a building warm shouldn't be much of a problem. The air tends to be hot (probably colder in winter, especially seeing the elevation of the city), and it doesn't conduct much heat anyway.
Solid ground, however, is a different matter altogether. The immediate surroundings of most of the building would keep a temperature of 4 degrees Celsius at any time, year round. Rock also conducts heat quite a bit better than air, so any heated object (say, a building) would quickly be cooled down. There is also sunlight up above, not so much down there except for a couple of hours around noon. Add to this the tendency of cold air to sink down, and the earthscraper would require a lot of heating to stay comfortably temperate.
All in all, the earthscraper sounds like a terribly impractical idea. I also wonder how it would fare, given Mexico City's problem with sinking ground water levels.