Your Dual Ethernet AirPort Base Station (M8209LL/A) is only capable of operating in the 802.11b radio mode. It also only supports WEP for wireless security and, unfortunately, it does not support WDS (which would be required to potentially extend your 802.11g Linksys router.) Basically, you will not be able to use it as you intend to.
To have some success in creating an extended network in your home, you may want to consider getting more up-to-date networking hardware ... Apple or not.
If I plug my ethernet cable (which extends to the far end of the house from my Lynksys wirekless router location) into the Apple base station and then use a LAN cable to my G4 (ethernet vs airport connection) , will this Airport base broadcast my signal at THAT end of the house so I can use the ipad there?? Right now I have the ethernet cable from the router plugged in to my main computer, the Powerbook G4.... But I would like to extend my wireless capabilities to the far end of the house. Or do I need an Apple Express? or something else? Model?
The Dual Ethernet AirPort is an outdated model so I'm not sure that it would work as you intend it to. However, it may be worth a try.
What you will be attempting to do is configure both the Linksys and the AirPort into a roaming network. The key to a roaming network is that the two routers must be interconnected by Ethernet. Note: The Eternet cable run should not exceed 300'.
Again I'm not sure that this will work with the 802.11b AirPort. If it does, please post back that you were successful setting this up. If not, then I would recommend that you get an 802.11n AirPort Express to be used in the roaming network configuration to replace the 802.11b AirPort.