How best to connect a switch to my Airport Extreme?

Just to clarify........the switch can be located anywhere you want as long as it connects to the AirPort Extreme. You can also connect a second switch to the AirPort Extreme if you need more ports, or connect a second switch to the first switch.

By "anywhere", you mean provided it's after the Airport/router?

Correct, as we previously noted that the AirPort must connect directly to the modem. Keeping that in mind, what we meant was that the switch does not need to be at the same location after the AirPort, it can be anywhere that you want after the AirPort.

A simple modem expect to only "see" one network device connected to it. Typically that would either be a single computer or a router. If you place an Ethernet switch between the modem and the router, any other device connected to the switch will also be "seen" by the modem ... and basically will confuse it. That is because, the modem will attempt to connect to the first device it "sees."

In your second scenario, you would effectively have four different devices vying to connect to the modem. Again, the modem would attempt to connect to the first device it can access. There would be no guarantee that device would be your AIrPort Extreme.

As Bob has mentioned, a router "knows" what devices are connected to it and will communicate with them as required on separate ports. This eliminates the issue with using an Ethernet switch. What may be confusing is that your AirPort Extreme has a built-in Ethernet switch, but because it performs as a router, it (again) knows what's connected to each of those switch ports. Also as Bob has mentioned, you can extend the number of ports by adding your Ethernet switch to one of the LAN-side (opposing arrows) ports on the Extreme.

Can this be solved using my existing gear......

No.

do I need to run a new cable?

Yes

Reason......a switch is not a router. Or, stated another way, a router must connect directly to a modem.