Don't worry about ambient light with this polymerization. MMA absorbs the fluorescent light in the room very poorly, and the TEMPO derivative that gives you the "living" polymerization characteristics is what matters most.
If you left the reaction for weeks maybe auto photopolymerization would matter.
As a subtlety here, there is some reaction gradient with styrene and MMA in a nitroxide mediated polymerization like this. I'd have to go look it up, but I think NMP is "happiest" with methacrylates, followed by styrenics, followed by acrylates. So in the start of the reaction the polymers will be enriched with methacrylate, and as it gets used up styrene will be more enriched. Its been many years since I ran the reaction, so maybe its vice versa. Anyways, shouldn't matter much for you.