Fastest way to cure a hangover; learned through experience:
In 2002 I was a Private in the US Army. We were in South Korea and were coming back to the States the following day. Our commander gave us a pass on the condition that we were not to drink. Well, being lower enlisted in the US Army, we drank. And drank. And drank. When we returned to base, our inebriation could not be disguised. The First Sergeant layed into us for about a half hour and then had us GI the latrines, laundry facility, and mess hall. After cleaning each one, unbeknownst to us, he came back through and trashed each one again. After several hours of this process, we all had moved on to the hang-over stage. When we had cleaned everything for the last time, the First Sergeant then decided it was time for a couple laps around the post. After the second lap, hang-over was gone completely!
Hang-overs are a result of lack of oxygen to the brain and exercise floods the brain with O2. So, even though it sucks, exercise!