How long it takes for the galaxy to orbit the univese?

Answer 1

The idea that the entire universe itself is spinning remains a question in astrophysics, and opposes the commonly accepted model that the universe is generally homogenous and isotropic - the same in all directions. Some studies hint that the universe itself may have a rotational motion but the concept has not gained general acceptance.

There are of course, orbital motions on smaller scales; rotations do seem to be part of a hierarchy ever increasing in scale; our sun taking about a fifth of a billion years to orbit the galactic center. It might be helpful to add that the galaxy is gravitationally bound to a group of galaxies forming a component of a larger scale galactic cluster which is in turn part of a supercluster; and these may be components of the largest scale structures known in the universe known as galactic filaments or walls; some of these structures appear to have intrinsic angular momentum around a common center of gravity, and, per laws of orbital mechanics, the further apart or bigger the structure, the slower it rotates.

Answer 2

Galaxies are not known to orbit the universe because the universe itself is expanding. Galaxies move through the universe due to the expansion of space, but they do not orbit it in a traditional sense.