AN UGLY CUSTOMER.--Our friend Billy Alexander, who, by-the-way, is quite a Nimrod, and the acknowledged champion shot in the country, killed a very large fat bear last week in the swamp back of "Middlesex Plantation" in this county. It's gross weight was about 700 pounds. The net weight 465 pounds. One half of it sent to Vicksburg, sold for $72. Although Billy did not seem inclined to boast of his feat in overcoming this monster single-handed, it was nevertheless an adventure which displayed a great deal of genuine courage, and would have covered us all up with glory.--Billy says that when he first discover "Bruin" his lordship was industriously engaged in raking up acorns, and had about four bushels gathered, and had his head turned from the direction of attack. Our hunter walked up within fifteen steps and gave the unsuspecting brute a charge of buckshot into his rear, which however only caused him to kick up like a mule, until doubtless, from curiosity, he turned to see the author of this ungentlemanly assault, when Billy gave him the contents of the other barrel of shot-gun. This had the effect of stunning him for a moment; but recovering, he made a desperate lunge at his persecutor. Billy was prepared, and administered another dose of lead, which made him yell most hideously, and maddened with pain, seemed bent on revenge. The fourth shot however, laid him out. We are under obligations for that 'bar' oil, Billy.