There are many ideas out there on ways to "cure" a hangover. So let's separate a little fact from fiction.
Can we sweat out the pain? Not exactly, but some fresh air and a light jog or brisk walk could help to reduce some of the grossness that is being felt. Just be careful not to overdue it, as dehydration is a major factor in hangovers and could feel worse when combined with intense exercise.
Pop a Tylenol and drink lots of water. Well drinking a lot of water is a good call, but Tylenol isn't the best bet given how it's metabolized by the body. If the pain is really bad, try Ibuprofen (advil) instead.
"Keep the party going" meaning to drink in the morning to avoid feeling the pains of the hangover. This really just delays the inevitable... the hangover will come and it will likely be worse.
Greasy breakfast food. While some claim this to be true, there's no evidence to support it. The reality is feeding the body a light and healthy breakfast with lots of liquids (alcohol-free) will be more beneficial in the end.
The truest way to cure a hangover is simply giving the body time to recover. Sleep when possible the following day, particularly since the night after drinking is typically not very restful. While sleep may come quickly, the actual restfulness for the body is minimal. Allow the body to recover, rest, and refill on liquids as needed.
The best solution is prevention either by abstaining or if a person chooses to drink they do so in moderation by staying in the blue zone. This means staying within a .02-.06 level of intoxication or about 1 alcoholic beverage an hour. Doing things like alternating water after each drink can help to reduce the hangover effects, as alcohol dehydrates the body and causes the hangovers to feel that much worse. By drinking more water throughout the evening, the body will be that much more hydrated. Moderation is key to a better, less painful morning.