After the initial response to my post, I mentally made a rude gesture and had a friend in the US get the download file for me.
Re-reading this thread, the link that Liam provided (the second reply was the one that made me throw up my hands and give up) is actually working. Why I am unable to google 'Adobe Air' and get the same thing (As that was my initial assumption, that I had an old link; so I tried Googling), I don't know.
I provided my buddy in Boston the same link I was trying and he was able to access it. Maybe it was coincidentally updated at the same time I was trying? Or perhaps Google hasn't updated yet; I suppose it's impossible to say.
In either case, the link Liam provided in the first reply (https://get.adobe.com/air/) is working for me as of this morning.
Good luck.