I only have a Cable modem with a single ethernet port and a Ethernet switch. Will a router work instead?
Yes, you need a router -- the switch by itself won't work. If your router has four LAN Ethernet ports (most do), just plug your PC into one of them and the OBi into another; the switch will not be needed. Of course, you also need to connect the WAN (Internet) port of the router to your cable modem.
Whenever you change the device connected to a cable modem, you need to power the modem off and on again, in order for it to recognize the new device.