Pull all the linens off the bed, and wash them as usual, using the hottest water setting your type of linen will tolerate. If in doubt, use a cold water setting and laundry soap. Dry as usual.
Sop up as much of the juice from the mattress using old towels or several layers of paper towels. Push the towels into the mattress using your body weight to pull up as much of the juice as possible. Continue doing this until no more moisture comes out onto the towels.
Take the mattress off the bed, and turn it on its side by propping it up against the wall. This prevents any excess fluids from seeping back into the mattress as you clean it.
Mix 1 tbsp. of liquid dish soap with 2 cups of cold water. Wet a sponge or clean rag, and dab the solution into the juice stain until the entire stain is moistened.
Dab at the stain with a clean towel or rag to remove the dish soap solution. Repeat until the towel is dry.
Continue wetting the stain with the dish soap and dabbing it dry until the entire stain is gone.
Rinse the stain with cold water on a sponge, and dry it by pressing towels into the mattress. The mattress must get as dry as possible. Leave it in front of an open window, or point a fan at it to help it dry completely.