You can do two things...
Drink PLENTY of non caffine/alcoholic/carbinated beverages.
These tend to "DRY" you out, but don't flush anything.
Thus.. Option one, drink LOTS of water or juice. This will "flush" the cold out. This will "flush" the cold out thru natural release, lots of trips to the toliet.
Drink lots of alcohol, and sweat it out. Yes, get trashed. But do it by drinking shots, dark alcohol works best.
Either way, the only way to "end" the common cold is to "flush" it out. You can't kill the virus, just get it out.
You can alway run a few miles, and get the same result, but most of the time, you don't feel like running 5 or so miles.