Does the air absorb the solar radiation and heat up, and then emit other energy and cool down? Or is something else going on? The answer, which will be a surprise to some of you, is that something else is going on. And understanding this particular facet of the diurnal temperature cycle is a key to accurate weather forecasting, especially when it comes to outforecasting the numerical models.
The air is nearly transparent to solar radiation. (Nothing shocking here; after all, it does get awfully bright outside during the daytime!) Therefore, unless it's blocked by clouds or haze, just about all of the radiation reaching the Earth's atmosphere penetrates down to the Earth's surface. A little bit gets absorbed by ozone, especially in the stratosphere, and a little bit gets absorbed by oxygen in the lower troposphere, but most of it makes it through to the Earth's surface. There, much of it does get absorbed, and this causes the temperature of the surface to rise. The air warms up because it is in contact with the Earth's surface. Air at the surface warms up, and as parcels of air get transported upward in thermals or turbulence, that heat gets distributed into the lower atmosphere.
In general, this heating or cooling only affects the air which is closest to the ground, roughly the lowest 1 km (0.6 mi) of the atmosphere. Above this, the temperature is almost steady, only varying by about two degrees Celsius between daytime and nighttime because of the weak absorption of radiation by the air.
So the chain of events is:
At night:
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