The title’s pretty much got it all.
May I just say: that screen is gorgeous. Where did you get it? Also, do you know if it’s depicting any place in particular?
As for the damage, my google-fu came up with this site: do-it-yourself-repair.
I’m not sure it’s what you’re looking for, but at least it’s a start. Good luck!
Again, beautiful screen. I want one 
Do you have any indication that the screen might be worth something? If it is, I would not trust anyone that isn’t skilled in restoration of such art – certainly including yourself.
Fakes are extremely common, unfortunately, so it is entirely possible that this work is not worth much.
I think the first step is finding out how much it’s worth so that you can decide what would be a reasonable amount to pay to have it repaired.
Autoclytus: you site shows how to repair shoji , which isn’t the same thing at all as what Ringo has: a byobu.
I had no intention of attempting a repair myself - rather I was wondering where one begins to look for an artisan skilled in this particular medium. Perhaps a museum is a good place to start asking.
All I know about it is that it’s thought to be about 100 years old and my father brought it back from Japan shortly after WWII. It was given to him by a Japanese in return for the use of his recording equipment.
I’d contact an auction house to get you in touch with a specialist in that area.(like the people that do appraisals and such on antiques roadshow. A company local to me is http://www.skinnerinc.com/ I’m sure you could send them an e-mail and ask.