I am trying to install a very old (circa 1920) telephone in my home. The cord from the phone has 2 leads, and the patch cord purchased from Radio Shack has 4 colored leads (red, green, yello, and black) on one end and a plug-in jack on the other. How do I install this old phone?

There's a good chance I'll be corrected here, but as I recall, you really can't. I actually tried to do this back in the 1950's and it has to do with voltage differences on a basic level and then just several more reasons beyond that. I think the transducers in a 1920's phone were some kind of carbon device and it just goes from there...
I've seen replica phones from that period and they All have modern electronics inside.

AltaVista may be correct in the answer above concerning the voltage differences. 1920 is a long time ago in the development of the telephone and electrical components and their design.

I was able to hook up an old phone in my home that I think was from the 1940's. There were only two wires. Also, from my experience in rewiring the entire phone system in my home (I have six phones) I have found that any of my phones work with using just two out of the four wires. I don't know why that is. You might want to try experimenting by hooking up combinations of the two wires from the phone with any two wires from the patch cord. Maybe one of the combinations will work.

If that doesn't work, maybe you could put the workings of a modern phone inside the old phone? At least it would still look the same then. Just an idea.

Are the wires twisted? If they are twisted pair, you should be able to hook it up. Even modern wired phones only need one pair. Since it is low voltage, it should not hurt to experiment some. The extra "pair" of wires is in case you have more than one phone line.

Modern single line phones always use the just the red and green wires. I am not sure how to sort out which wire from the phone to connect to the red and green. I don't know if you opened the phones up if there would be enough similarities to check things like which wire went to the ringer, the switch, the speaker, or microphone.

Won't hurt to try. As indicated, connect the two wires one to red the other to green. If that doesn't work reverse them. If it still doesn't work, display the phone as a museum piece.