Wow I just had the same scenario today! Found out WCIII was updated for UB in January today so decided to gome home on lunch and find my discs. Had everything except the TFT case with the number. Luckily I found it under my car seat after 20m of tearing my room to shreds
Enough rambling.. no without the cd key on the case you will have issues with Battlenet. It is nowhere else by default.
Just go buy a new one. It's diiiirrrt cheap. An alternative is to check out your Windows registry (if your on a PC). Not sure if you can do anything the same on a Mac