How many days will the sun pass through the zenith in a full year?

Answer 1

If you are located anywhere on earth that's more than 23.5 degrees from the

equator, then the sun can never appear at your zenith.

If you're anywhere within 23.5 degrees of the equator, then the sun will appear

at your zenith, or very close to it, twice each year.

Answer 2

The sun passes through the zenith twice a year at each tropic line. This will occur once at the Tropic of Cancer (23.5°N) and once at the Tropic of Capricorn (23.5°S). So, in a full year, the sun will pass through the zenith two times.