You will probably find you want a motor + gearbox to increase the torque and reduce the speed, otherwise you will be limited to using a tiny pulley which will make it harder to stop the windings getting tangled up on the pulley.
I'm not sure whether you're looking for free DIY options to attach the pulley, or nice robust off-the-shelf solutions. For the DIY approach I would look for a spindle that is a similar diameter to the motor/gearbox output shaft, and connect using a plastic sleeve that is a tight push fit on both parts. Support the spindle on two bearings (loose fit holes through a couple of bits of wood should do the trick). To attach a pulley to the shaft, just wrap adhesive tape tightly around the shaft to build it up until the pulley is a tight push fit.
The 'off the shelf' approach would use a universal coupling between the motor/gearbox and the shaft, and a boss secured to the shaft with a grub screw (worn out drive pinions are excellent for that) with the pulley bonded to the boss - or buy a pulley designed to mount on a shaft.
The wooden sewing reel you showed looks like a good basis for the pulley, but make sure you use suitably stout cord - think shoelace rather than sewing thread - so it doesn't wear out too quickly. (This will have implications for the size of the pulley you need).