Use Latisse As an Eye Drop?

You definitely need to clean the eye before applying the latisse. This is also why Allergan has given multiple brushes, so that a clean brush is used with each application.
Do not place the latisse directly into the eye. Latisse is actually a glaucoma medication. It is sold as Lumigan and this is dropped into the eye. This helps control intra ocular pressure. if you do not have glaucoma you do not want this in the eye just on the eyelid.
I would manage the sty with warm soaks. the majority of these will resolve on their own. If it progresses do not hesitate and get to a doctor for antibiotics.
All the best

 Latisse should not be used as an eye drop. Latisse is a prescription-only solution that has been proven to stimulate the growth of eyelashes. Within two to four months, your eyelashes can become longer and thicker. Latisse is safe and is the only FDA approved treatment for growing eyelashes. Latisse is applied daily with a disposable brush which is included with product. Latisse is gently applied along the base of the upper eyelashes. It may have been coincidental that you developed a sty but if this problem recurs, you can use Latisse in the morning before applying makeup.