Wiring a 1920s intercom to a contemporary door-entry phone
I may be a year to late, but yes you may use it, and in fact your "new" wooden phone are more advanced than the plastic one.

Somehow you need to configure the hook switch with 2 sets of contacts or 3  contacts opening when going on hook. 
The impedance of your receiver may not match the one in the other phone, in that case we must try your induction coil/transformer, but you will probably make it good without, just copying the diagram you showed. The old transmitter may be another thing, the carbon granulates may be stuck in there, and some shaking, or even a little knocking on the rim of the transmitter element may help. The ringer should be fine as it is. You will find dedicated expertise on the Rotary phone forum.

http://www.classicrotaryphones.com/forum