Cindy,
I don't see anyone answering your request but I'll comment briefly. The fence in your link looks like a simple cedar panel fence with a 'roof'. It probably has a Japanese name but I don't have any references in front of me currently so can't give you a term (where's Yama/Mike when we need him?) This would be easy to make, but if you buy individual panels pre-made like this the cost will be prohibitive.

If you want to go the pre-built route, you can get some excellent fence ideas from one of my favorite wood sources, which offers many types of bamboo fencing, with or without cedar frames (i.e., these *really* look Japanese :-)).
I've posted the link below, check it out. Bamboo is even better than the pic you were looking at, IMHO.

Here is a link that might be useful: Japanese fencing supplies