Most brown recluse bites heal just fine without any medical intervention. The first thing you should do for a new bug bite is wash it with mild soap and water.
From there, you can apply simple first aid to ease pain and swelling. The RICE method is recommended.
RICE stands for:
Start by icing the bite area, taking care not to apply the ice directly to your skin. If possible, wrap the area with a compression bandage and then elevate it above the level of your heart.
Any time you place a bandage over a wound, make sure to remove it at least once per day to check for signs of developing infection. After removing the bandage, clean the wound with soap and water again, pat it dry, then re-dress it.