Assuming precession occurs on a 26000 year period, will we eventually have to change time zones (forward or backward) to adjust for this? The way I see it, in continuous time, we will eventually be at a different point on the axis, thus having less or more sun during a day. Am I missing something?
I believe the source of your confusion is that you are confusing a sidereal year with a tropical year. The sidereal year, which is the definition most often cited as the standard definition of a year, is not the year our calendars use. That year is the tropical year. The sidereal year is the time it takes for the Earth to complete one orbit around the sun (sounds familiar, right?). The tropical year, the one we actually use to keep time, is the time it takes for the Sun to complete its apparent journey from one polarly extreme to the next (e.g. summer solstice to summer solstice). The two differ by about 20 minutes. In about 13,000 years the constellations we see in the nighttime sky in June will be the ones we see in the nighttime sky in December, but the sun will still be above you (more or less) at noon on that day.