thing sin the universe spin because they contract. When everything was all spread out everything had random movement. So it mostly averaged out to no spin but there would be a **very** small net spin.
But as things collapsed into planets, starts, and galaxies, they contracted. As their mass comes closer to their center of masses angular speed will increase. It is the same effect as when a skater tucks in their arms to spin faster.
Since the universe as a whole is expanding, and it is only its constituent parts that contract (read the average distance between everything is increasing) one can logically conclude the universe is not only not spinning in any appreciable sense but in fact what inperceivably small spin it did have has been forever slowing down.