What is the orbital period of our sun around its barycenter? In years, what is the time it takes our sun to orbit around our solar system's barycenter?
As you can see from the diagrams below of the motion of the Solar System's barycenter relative to the Sun (courtesy of Wikipedia), 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.

(Click the above image for a SVG version).

It is primarily dominated by Jupiter's period (11.862 years) because Jupiter has most of the non-solar mass of the Solar System. (On a related note, Jupiter has most of the angular momentum of the solar system). Jupiter's mass is approximately 0.0009543 solar masses, so the Sun is almost 1048 times more massive than Jupiter.
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.


Here is a Sage / Python script which displays an interactive 3D plot of the above diagrams, using data from
the JPL Horizons on-line solar system data and ephemeris computation service. Horizons can produce data covering a wide range of dates, please see the Horizons documentation for details.