In years, what is the time it takes our sun to orbit around our solar system's barycenter?
As you can see from this diagram of the motion of the solar system's barycenter relative to the Sun, from 1945 to 1995 (courtesy of Wikipedia)
(click here for a SVG version)
the Sun's motion with respect to the system's barycenter is not a simple closed curve, and it doesn't have a simple period.
It is primarily dominated by Jupiter's period (11.862 years) because Jupiter has most of the angular momentum of the solar system.
According to Wikipedia
To calculate the actual motion of the Sun, you only need to consider the motions of the four giant planets (Jupiter, Saturn, Uranus, Neptune). The contributions of all other planets, dwarf planets, etc. are negligible.