One thing you've also missed out in your diagram is the centrifugal switch shown in the picture of the stator coils, this opens the start winding when the motor reaches 75-80% of full load speed, ie: the start winding is never permanently energised.
As far as sizing a suitable start capacitor goes, generally 50uF is a suitable starting point.
As a note of caution, be VERY careful with energising really old gear like this, that switch looks like it's come to the end of it's life, and I would suspect that the transformer would require a megger test before it was livened again.
Be Careful!
