Rinse duck under cold water and pat dry. Remove any large lumps of fat from skin. Rub inside cavity and skin of duck with lemon. Season cavity and skin with salt and pepper. Truss bird, place on rack and set in shallow roasting pan. Roast at 350°F for 30 minutes.
Pierce skin all over with fork to release fat. Continue roasting until duck is completely cooked, around 1-1/2 to 2 hours.
Once every hour, remove all but 1 cup of fat from pan (some remaining fat will keep pan from scorching). Increase oven temperature to 500°F for the last 15 minutes of roasting to crisp skin. Remove duck from oven and set on warm plate. Pour off and discard fat.
Place roasting pan on top of stove. Add port and heat over low heat, scrapping up any browned bits. Add butter and heat until melted. Add flour, stirring until smooth. Add chicken broth and whisk constantly until smooth and thickened, around 2 to 3 minutes. Stir in orange juice and 2 tablespoons brown sugar and mix well. Add cranberries and cook over high heat until cranberries pop, around 2 to 3 minutes. Add cayenne and more salt and pepper to taste. If mixture is too tart, add more brown sugar.
To serve, using poultry shears cut duck in half lengthwise. Place on two serving dishes. Pour half of sauce over each serving. Serve immediately.