How to Reupholster Arm chair?

Get a book on upholstering furniture from the library. Frankly, if you're just reupholstering one chair, you'd spend so much money on basic tools and materials that you'd probably be better off having the work done by a professional. 

Your cost figure for reupholstering a chair, any chair, is much too low. Perhaps that is just an estimate for an hourly labor cost for professional work. You have to figure the cost of the work by including how many hours the professional would work, plus the cost of materials. You could save money doing the work yourself, but without the right tools and a realistic idea of what you're doing, the finished result may not be a nice as you would like.

The instructions and advice on how to re-upholster furniture are too long to write out here. I suggest looking online for further instructions, or find a book that explains it all

Most upholstery leather is fake. It is a bonded material that eventually starts peeling and there is not much to stop that happening. I would change the covering rather than repair it. Leather is difficult to use if you have never done any upholstery, but fabric is much easier for a beginner and is not much more than the cost of the material.