So I gave in and tried calling the number, and it was indeed the SWISS International first class line. I'm not sure what you got, NYC1, but it worked for me, it was just a very expensive call because I did a straight-forward call from my US mobile (vs. Skype or Google), since I figured I would get either nothing or a recording.

I was prompted to press 3 for English, I pressed 3, and the call was picked up immediately. I asked the man if this was a general reservations line or a first class line, and he said that it was for first class customers.

SO...I guess others' mileage may vary? But the number I posted above (+41 848 365 365) worked for me for SWISS first class reservations.