Why is it colder at the top of a mountain if it is closer to the sun?

The sun is about 93 million miles away, so climbing even 29,000 feet closes the gap by a fraction of a percent far too small to matter. What actually governs mountain temperature is air pressure. Atmospheric pressure drops with altitude, and when a parcel of air rises into lower pressure it expands, and expanding gas cools. This is adiabatic cooling, meaning it happens without heat being removed, purely through the work of expansion. The rate is roughly 3.5 degrees Fahrenheit per thousand feet in dry air, closer to 2.7 when moisture is condensing and releasing latent heat.

There is a second reason that matters as much: the atmosphere is not heated much by sunlight passing through it. Sunlight mostly passes through air and is absorbed by the ground, which then warms the air above it through conduction, convection, and infrared radiation. So the air is heated from below, and the farther you get from that heat source the colder it becomes. Thinner air at altitude also holds less heat and radiates it away faster, which is why mountain temperatures swing so violently between sun and shade and between day and night. The same physics explains why the summit of Kilimanjaro carries glaciers three degrees from the equator.