Bugh graduated from Indiana University law school, moved to Wisconsin and eventually settled in Berlin where he practiced law and was the editor of the local newspaper. During the Civil War, he served in the 5th Wisconsin Volunteer Infantry Regiment and was badly wounded. In 1866, he served in the Wisconsin State Assembly. He was then appointed postmaster of Berlin serving until his death.

"Case 137. — Captain William A. Bugh, Co. G, 5th Wisconsin, aged 35 years, was wounded at the engagement at Williamsburg, Virginia, on May 5, 1862, and after lying a few hours on the field he was removed to a temporary hospital, and thence to a hospital transport in the York River and sent to Baltimore, where he was received at the Camden street U. S. A. General Hospital on May 10, 1862. A conoidal musket ball had entered the right groin, passed slightly downward, traversed the line of union between thigh and trunk, fractured the neck of the femur in its transit, and emerged posteriorly at the fold of the buttock. On flexing or rotating the thigh crepitus was plainly distinguished. His limb was suspended by Smith's anterior splint, and this treatment was continued for two months. The case progressed without a single untoward symptom, and in the middle of July, 1862, consolidation of the fracture was sufficiently firm to permit the patient's removal to the house of a friend. The limb was shortened one and a half inches. In October Captain Bugh was able to move about on crutches, and the wounds were entirely healed. About this time he took a journey to Washington, and was promoted to a lieutenant colonelcy in the 33d Wisconsin and placed on recruiting service. He served until April 25, 1863. His recovery was so rapid and uninterrupted that he reluctantly assented to the opinion of his surgeon, Dr. Edmund G. Waters, that he would be incapable of active duty in the field. A letter was received from Lieutenant Colonel Bugh, dated June 12, 1867, more than five years subsequent to his injury, in which he stated that he had partial anchylosis of the hip joint and was unable to perform any labor in a stooping posture. Otherwise his condition was satisfactory, though he was more readily fatigued and debilitated than before he was wounded. He suffered no inconvenience from the slight shortening of the femur. His death occurred eight years afterwards, on August 19, 1875, of phthisis pulmonalis. Examining Surgeon N. M. Dodson, of Berlin, Wisconsin, informed this office on December 13, 1878, that " no post-mortem was had in the case of Captain Bugh. The wound healed in 1866, and the limb remained sound afterwards. Bugh filled the position of postmaster, was much upon his feet, and residing for some years three-quarters of a mile from the post office, he walked to and fro three times a day with very little inconvenience, using only a cane. The joint was firmly anchylosed. He suffered from pain of a neuralgic character in the whole limb, and at night frequently with cramps in leg and foot." -- The Medical and Surgical History of the War of the Rebellion. Part III, Volume II. (3rd Surgical volume) by U. S. Army Surgeon General's Office, 1883.