Welcome to BYC!
What a lovely barn and pasture!
What is your purpose for covering to help determine the "best" way? Predator protection, weather protection, bird containment, etc? How much is cost a factor, and what skills are ya bringing to the table?
Covered netting may help with aerial predators but it won't do anything for weather.. like baking hot sun, pouring rain (sheeting off the roof into the area), snow. etc. It's hard to imagine anyone not facing at least one of those conditions each year.
I built (uh supervised

) a lean to wooden structure and covered with corrugated roofing panels.. metal ones by my barn and clear or plastic ones where more light transmission or lighter weight is preferred, or even over just a portion.. leaving half with the netting and half *more* weather resistant.
We were able to use the existing framing in the pole barn to attach the "lean to".
If using soft netting.. I would suggest using something heavy duty and UV protected.. that "bird" garden mesh is crud.. breaking easily, catching and killing song birds often, and leaving lot's of decaying broken pieces around for birds to ingest.
Hope this helps to ask some of the right questions!
Actually.. looks like you may have a similar lean to portion like I'm describing on the back of your barn already (they're almost twins).. So ya, ultimately knowing your long term goal would help (me anyways) as far as what I might suggest or not. It was important for us to still be able to drive an rv height trailer past the side of the barn.. so any consideration (for us) would have to be either higher than that or semi easily removable if needed.
Re.. securing it on the barn side.. we attached 2x4 on the outside with long screws to the wood framing on the inside.. and easily attach most anything desired to that.. Or even just screw directly into the inside wood framing from the outside (through the metal, using gasket screws if desired).. but "hanging" stuff never works out well long term in
my experience. Also I would probably go a little higher up on the barn wall side so the fence post was the lowest point.
Best wishes and happy adventures!