This really depends on the the additive.
Polymers generally all come as white/colorless. If they are in their pure form, the are rather unstable. This is because UV light initiates a free radical oxidation step. Some of the symptoms of this step include color change and enfeeblement. Check out how yellow this historical sample of PE got in the past 80 years:
As we started producing plastics more often, we started including other chemicals in the melt called stabilizers. We include these chemicals which are designed to absorb UV photons that would affect the polymer. These chemicals are usually rather large "small molecules" with lots of conjugated loops double carbon bonds. Here is one that I picked blindly from google images:
For ABS, which is unstable over the ranges of 300nm-310nm and 370nm-385nm, we need to add something to mask these wavelength regions. So we add chemicals that mask these regions. I led up to this with the long intro because it is important. The colorant used in the polymer does not necessarily block UV light. Say you have a plastic with a red additive with peak absorbance at 633nm. That would probably not block UV photons.
So, back to your question. Is black plastic immune to this problem?
Probably. Most black plastic is made from adding carbon black to the sample. The carbon black absorbs over a wide range of wavelengths. There is a simple white paper about this
here. We can assume that if the plastic is made with carbon black, then it is built to last in the sun. I hedge my answer by saying "probably" because without some in depth analysis, we can't tell if the plastic was made with carbon black or something else.
I hope that is a satisfactory answer to your question.