Some time ago I bought a detached house that also had a carport attached to it.
The problem here is that the side of the carport faces the neighbor. The 1st photo shows the view down from the carport. The wooden beams you can see there protrude a bit, so rainwater is constantly running off onto the wooden beams and fence, so they are already relatively badly damaged.
What is the easiest and cheapest thing to do in a case like this?
The wood gets wet and dries out again. Other than regular painting or oiling, there is nothing you can do.
I wouldn't worry about bearing capacity. A wood shed or timber frame stands forever and is exposed to the weather 24/7.