Will my bamboo fence discolor and will it last in winter?

Bamboo is a woody plant. The same things that attack wood attack bamboo as well. Generally the winter is not a problem. Cold preserves things. Water and sunlight are the enemies as with any bare wood outdoors. Rain is generally not a problem unless the wooden structure cannot drain and get dry enough to stop rot-producing organisms from thriving. Do not put bamboo in the ground unless treated. The UV in the sun will attack the natural wax in the bamboo skin. The result is a weathering to a silvery gray, just like cedar. You can greatly retard this with a waterproofing sealer as is used on wooden decks. I like Cabot’s best. The installation should allow the rainwater to drain off.