Calculate the time when Arcturus passed through the meridian
Find the time that Arcturus passed the meridian in 29 April.


With respect to the Sun, the Earth rotates once every 24 hours.  But because the Sun's apparent position changes in the sky due to Earth's revolution around the Sun, the Earth rotates once every 23 hours, 56 minutes, 4 seconds with respect to the stars.  Because of this 3 minutes, 56 seconds difference, all stars rise and set 3 minutes, 56 seconds earlier each succeeding day.  This applies to the time of a star passing through the meridian as well.

So, on the next day, April 29, Arcturus will pass the meridian at 22:56:04.


  And what day does the Arcturus will pass the meridian about 01:00:00?


Over the course of an entire year, about 365 days and 6 hours, these 3 minutes and 56 seconds adds up to an entire day, so that exactly one year later, the same star will pass through the meridian at the same time of day.

For Arcturus to pass the meridian exactly 22 hours earlier, it should be 22/24 of a year before then, or about 335 days later, or about 30 days earlier, or March 29.


  solar time equal sidereal time minus 12hr


Generally not correct.  Solar time is based on the average time it takes for the sun to cross the meridian (24 hours), and sidereal time is based on the time it takes for a star to cross the meridian.  Sidereal time counts 24 "hours" in 23 hours, 56 minutes, and 4 seconds, meaning that it runs slightly faster than our normal solar time.

There is one day in which solar and sidereal time are 12 hours off, on the northward equinox in March.  The Sun is on the northward equinox, but solar time is measured since midnight where sidereal time is measured since passing the meridian (local noon).

There is one day in which solar and sidereal time coincide, on the southward equinox in September.  The Sun is on the southward equinox, 180 degrees in the sky from the northward equinox, so the 12 hours off from midnight and the Sun being 12 hours off from the northward equinox cancel.