The details will differ depending on the type of lead screw you use.
For allthread lead screws, you will need the 1/2" square nut, 1/2" allthread lead screw, two 1/4" screws and nuts to hold the square nut in place, two 1/2" ID bearings, two standard 1/2" nuts, and one coupling hub.
Attach the square nut to the nut support using the two 1/4" screw and nuts. The bearings will need to be inserted into the seats of the lower and upper part of the z-axis. Insert the lead screw through the top bearing. Use one of the standard 1/2" nuts and start threading it on the allthread screw just below the bearing. Lower the lead screw to the square nut and start to screw the lead screw into the square nut. Continue until the screw is near the lower bearing. Add another standard 1/2" nut to the screw and keep turning the screw until the screw just passes the lower bearing. Add the coupling hub to the top of the screw just above the upper bearing. Turn the lower and upper standard 1/2" nuts until they are snug against the bearing. If the standard 1/2" nuts become loosened, consider adding another nut to each end against the existing nut to keep them in place.
For 1/2" acme 5 start lead screws, you will need to use an antibacklash nut in place of the square nut. This nut attaches with a flange using two #8 screws and nuts. The standard 1/2" nuts are replaced by clamping collars.
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I couldn't find any information about how to mount the "z-axis lead screw in the book" (Build Your Own CNC)