Cleaning a syringe with alchohol is a bad idea as it tends to dissolve the rubber plunger inside, making the seal not fit properly. Try hydrogen peroxide, which studies show is a better sterilizing agent anyway. It has no corrosive properties like alchohol.
That's assuming that you actually want to do this and are not able to pony up the 10cents for a new one, or wait the couple days it will take to arrive.