Does the universe prefer a velocity? As I understand it, the distribution of matter is roughly homogeneous across space -- anywhere I stand in the universe, there will about the same amount of matter in every direction around me.

Is the distribution of matter across velocities roughly homogeneous?  If I flew away from the center of the Milky Way at .999c, would all directions look roughly the same, or would I notice a wind of galaxies and neutrinos traveling at a preferred velocity?

Tags I cannot add: cosmological principle, homogeneity, isotropy
There is a preferred standard of rest, that of the Cosmic Microwave Background (CMB). You can detect motion with respect to the microwave background. So move fast with respect to the background and the CMB in the direction of motion will be blue-shifted and in the opposite direction will be red-shifted compared to a direction orthogonal to the direction of motion.

The Earth motion has been measured with respect to the CMB, see here

You could/can also detect your motion from the anisotopy of galactic red-shift in the same way.