Does angular velocity have an upper bound?

The answer to your question is yes, it puts an upper bound on how fast you can rotate an object.

Now, I suppose someone's next questions is likely to be "well, what's the fastest angular speed at which an object can rotate?" The answer to this is that it depends on the size of the object. According to the most basic classical/relativistic analysis, no point on the surface of a rotating object can move tangentially faster than the speed of light. In other words, the fastest angular speed of a rotating object is:

$$\frac{c}{R} = \omega$$

While we're on this subject, this very argument was the one of the first clues that the electron's "spin" was not in virtue of any literal spinning.