What port(s) does Apple Continuity feature on iOS 8 and Yosemite use?

IIt probably isn't using any ports at all. Continuity is embedded deep within the Bluetooth and WiFi hardware. There are only a couple of specific operations that eventually wind up going through something recognizable as a TCP/IP port.

OK, but I have the problem that when connected to my work wifi, continuity doesn't function. The iPhone and Macbook are close, and both connected to the same wifi SSID. At home it works as advertised, and I always assumed the difference was the aggressive firewall they use at my work. If they directly connected over WiFi, then it would completely bypass the network hub at work, and there would be no difference, right? And if they did directly connect over Wifi, why did Apple create the restriction that they share the same wifi access point? Security?

Is that the same MacBook that you have at home? No one really knows the details of how Continuity works. It may very well not work in an enterprise configuration. Yosemite itself barely works in that configuration. But many, though perhaps not all, of the communication actually bypasses your network infrastructure. It goes directly from device to device - but not always. If it works, great! If it doesn't, I wouldn't bother with it, because it probably never will with your set of devices.

Speaking of which, I went back and re-read your original post. Continuity most definitely won't work with an iPhone 4S. It will only work with computers made since 2012 and devices made since 2013. So maybe try with just the iPad.

I finally got this to function at work by using a different wifi network here.

And I also use an iPhone 4S. Handoff for phone calls (over cellular) function on the 4S, but not the other continuity features. The easiest way to determine if it is functioning is to try call a phone number from FaceTime on OS X.