There is no strict formula for designing a Japanese-style arched arbor, but there are guidelines that will help you recreate the aesthetic. A low, broad profile is one characteristic to emulate. Use a form that arches up a maximum of 1 foot in height for an arbor 6 feet wide is a good rule of thumb. The ends of the arbor arches should curve upward to a end in a point or a rounded shape, in keeping with the tradition.
Wood is the material of choice for a Japanese arbor. It can be left bare to weather naturally or painted red or black, if it's part of an overall Japanese garden design using traditional colors. Use smooth, sanded wood with softened corners rather than the rough-sawn lumber often sold for outdoor use. If the wood will be left bare, use a rot-resistant lumber, such as cedar or redwood, both of which have a reddish color.
You'll need a power sander, circular saw and jigsaw to build a Japanese arbor. In traditional Japanese architecture, metal fasteners are not used -- instead the wood is cut in ways that allow the joints to lock together, forming a sturdy structure. But if you're not a master carpenter, using a drill and conventional nails, screws or bolts is fine. Two-by-twelve boards can be used to form the arch. Draw the shape on the wood use a jigsaw to cut it out.
Part of what gives an arbor a Japanese look is how it is used as part of an overall garden design. In a Japanese garden, an arbor entry is typically built over a gate through a fence enclosing the garden. The fence can be covered in a little "roof" and designed together with the arbor as one seamless feature. Using sand as a ground cover, small boulders placed as garden accents, Japanese lantern-style path lights and traditional Japanese symbols can complement the arbor as part of the overall design.