WASHINGTON—After seeing exports to China tumble, U.S. farmers and ranchers are now bracing for more losses in their next-biggest Asian market: Japan.

On Dec. 30, Tokyo will begin cutting tariffs and easing quotas on products sold by some of American agriculture’s biggest competitors—including Canada, Australia, New Zealand and Chile—as part of the new 11-member Comprehensive and Progressive Agreement for Trans-Pacific Partnership.

The...