How do I know whether a circuit (originally for 5 wire) will work for four wire stepper motor?
Your circuit is for a standard four-coil (five or six wires) unipolar stepper motor. Such a circuit can cause current in one direction through four coils, thus making the four steps. Such a circuit has four open-collector/open-drain outputs.

A standard two-coil (four wires) stepper requires a driver circuit that can cause current through the two coils in two dircetions, thus again making four steps. Your circuit can't do that, you need a circuit with two full H-bridges (four push-pull output stages).