Good luck on your quest.
Jim
How long before a "newbie" goes "solo" depends upon how fast they learn and develop the skills. Everyone makes mistakes, but as a trainer, you need to feel confident they will not trash a print, damage equipment, have a xenon lamp explode in their face, or electrocute themselves because of incompetence, stupidity, or negligence.
Here are some links to training materials:
http://www.kodak.com/US/en/motion/screencheck/offer.shtml
http://www.kodak.com/US/en/motion/newsletters/index.shtml
http://www.smpte.org/smpte_store/books/
http://www.bksts.com/courses.htm
http://www.bksts.com/training/project.htm
http://www.bksts.com/Images.PDF
http://www.bksts.com/cinetec.htm
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John P. Pytlak, Senior Technical Specialist
Worldwide Technical Services, Entertainment Imaging
Eastman Kodak Company
Research Labs, Building 69, Room 7419
Rochester, New York, 14650-1922 USA
Tel: 716-477-5325 Cell: 716-781-4036 Fax: 716-722-7243
E-Mail: john.pytlak@kodak.com
Web site: http://www.kodak.com/go/motion
John Pytlak is correct. How long depends on how well he trainee learns. When I started back in 1970 the law in my province of New Brunswick required 500 hours of training. This was on the job, through a union, and there were no manuals at all. We learned from the operator in the booth. Thankfully times have changed and there is a wealth of valuable information on the WWW as well as from the SMPTE and Kodak. Send to Kodak for their projectionist training information. I believe it is free and it is great. I used it recently while training two people.
Approach this as if you were a teacher. You know what is needed to do the job so put down on paper all of the things that a projectionist does and set a schedule : ie: week 1..film handling: week 2..presentation (show them a good screen image and let them see what a good presentation looks like) etc.
If you know your job and they spend time with you then they will learn. The most important quality that a trainee needs is a desire to present a good film for the audience to see. Forcing someone who is not interested in film projection is not a good idea. Their lack of interest will soon show up on the screen.
Fiallly,this site is a resource. Great people who will always answer a question for you.
Good luck
Carl King
Also, it will depend on what you want the projectionist to be able to do. When I started, operators were expected to know how to change "accessable" parts on the projectors and assocated booth equipment (sprockets, gate springs, belts, diodes in power supplies, plug-in relays, etc.) If your operators only need to know how to "operate," that is, threading, make-up and break-down, recovering from a film break, the training time will be less.
Do not expect the student to do very well if you rush a training session between shows, then require them to go back to being an usher, concession attendent, etc. for the duration of the shows. Don't expect them to do well if you try showing them the box office, cashing out, concession stand, ushering and the booth all at once. Of course, you want the person to get up to speed. So a suggestion might be to concentrate only on threading correctly and fixing common errors, since that's what you need the most. Then, as the person gets used to handling film (and proves themselves), work on make-ups, changing bulbs, etc. Nothing's worse than spending a lot of time on a person, then having them quit.
When I trained, I went to the theater from 6:00pm to closing (12:00) for 4-5 days a week for about 3 months (single screen w/ changeovers and carbon arc.) No pay!
Part of this answer must depend on what the apprentice will receive in return, training for a job that may disappear within a few years.
As far as mechanical ability or knowledge of physics are concerned, it has been my experience that those who really want to know, find out for themselves.
It would be uneconomical for him/her to spend their time on the standards the older ones of us had to know to get a license in the old days; Ohm's law, lens calculations, ect. (Never knew a projectionist who was asked to use these anyways - still an understanding of the basic principles is never a bad thing).
I suppose the hardest thing to detect is the most valuable to you - the indefinable inborn quality known as "showmanship". The desire to put on the best performance you can, and to take pride in this invisible art. It is invisible because if you are doing your job right, the audience becomes totally unaware that you exist. A large part of this desire, will come from their work with you. The student must, at first, try to emulate the teacher.
In the brave new world that I live in since the disappearance of professional projectionists, by and large, I see people who just do this because the company pays them an extra dollar an hour to "learn the booth", and some who are genuinely interested in what the audience sees. All to often a "buck-an-hour" projectionist is worth exactly that.
Fortunaltely, we have a series of well-written, easy -to-read articles from Mr. Pytlak to help these people along. "The Seven Deadly Sins of Projection" should be a must-read for any apprentice, (and a read-again for all of us).
I have had the good fortune to train a dozen or so apprentice projectionists in my time. Eight of these passed the test, the others I talked into another profession. Of the eight who passed, none have ever given me any reason to be ashamed.
Unfortunately, most have them have left voluntarily or been forced, through circumstances beyond their control, to leave the booth.
Remind them often of the Lawrence Olivier line on performance, "Always strive to do your best. You can do no better. You must do no less."
Thanks. 
Here's a link to the article:
http://www.kodak.com/US/en/motion/newsletters/notes/june2001/pytlak.shtml
Please let me know any other topics you would like me to write about in future issues of Kodak Cinema Notes.
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John P. Pytlak, Senior Technical Specialist
Worldwide Technical Services, Entertainment Imaging
Eastman Kodak Company
Research Labs, Building 69, Room 7419
Rochester, New York, 14650-1922 USA
Tel: 716-477-5325 Cell: 716-781-4036 Fax: 716-722-7243
E-Mail: john.pytlak@kodak.com
Web site: http://www.kodak.com/go/motion
sorry shop hic