Actually there are several routers available which have a serial port connection designed to allow a dialup backup connection, though Linksys doesn't have one. Since a serial modem is hardware based, specific brand drivers aren't needed, as all modems are designed to respond to the standard Hayes command set.
In order to use a dialup as a backup connection without a router with a serial port, you'd need to set up a computer running Internet Connection Sharing (or a non-Microsoft equivalent) with a dialup connection configured, connected to your network with a network card.
When your cable modem is down, the easiest thing to do would be to log into the router, change its IP address to something besides the default; then boot the ICS computer, and configure ICS to use that original IP address for the internal port. That way the other computers on your network can simply start sending their traffic through that computer instead of the router. You may also need to turn off the DHCP server in the router and turn it on in the ICS configuration, since you may need to reboot the other machines to make traffic pass properly.
Alternatively, if all your computers connect to a stand-alone hub or switch and then the router connects to that device (for example using a router with only one LAN port), you can simply turn off the router entirely when the cable modem goes down, since it won't be needed.
When the cable works again, you just shut down the ICS computer, and set the router back to the original settings.
Oh, and RJ45 is 8 pins, not 6.