The scheduled puck drop time is available somewhere - our Bruins GDTs usually have it. Whether that aligns with reality or not (anthem singer holds on to the word "brave" an extra 40 seconds), who knows.
That said, I've found puck drops fairly consistent with respect to the arenas they play in. I imagine each arena has its own pregame show, and some might take 1 or 2 minutes longer. Bruins usually start about 7-8 minutes past the hour, unless it's a national game (either nation).
Playoffs, it's a whole different story. It's usually 15 minutes to open a series, and of course the mandatory 15+ minutes if your game is on HNIC.