With this tree nowhere near healthy, you're a ways from knowing what its growth habits are. You can definitively call it a shimpaku, but beyond that, I'm not sure anyone could say. Get the health of the tree sorted before worrying about airlayering or grafting. Your chance of success improves drastically on a healthy tree.
You have two paths forward. 1. Leave it alone until its health[y|ier]. Situate it in full sun and water it often. 2. Do a light cleaning, focusing on foliage that's growing/hanging downward. Leave the tips of branches alone! Wire primaries out/down. As you see new growth and vigor return, finish cleaning it. Of the two, I think the 2nd would have the quicker turn around. The 1st will work, but requires patience. Don't fertilize until you see lots more growth. Repot it in spring and make sure its immobilized in the pot.
I'm not convinced the poor growth is insect or fungal -related, but it would be something I'd monitor.
Do you have any professionals or advanced club members available to help you with the work?