Like I think I've posted before, it's the hormone cortisol that can really wreak havoc on some anxiety sufferers. It's produced by the body in the mornings to help us wake up. For me, oddly enough, it doesn't hit me right when I open my eyes. Only if I was having a weird or bad dream. Usually it takes all of one minute to kick in. Pretty much it hits me when I stand up.
I don't wake up anxious every morning, but often enough that I just consider it routine. Unless the anxiety is so bad that it affects my stomach I just work or do whatever I normally do. In about an hour the anxiety will go away. If it's particularly intense I will write, which is my preferred method of dealing with anxiety in general.
Sometimes the anxiety hangs on for a while but isn't as intense. Just annoying. At that point the best thing I can do is leave my apartment, and generally that will make it go away.