How Do I Get Rid of a Small Cyst on my Nose?

Hard to tell what is going on without a photo or exam.  You mention a nose ring.  It is possible that you have what is called an inclusion cyst, where cells from the outer part of the skin get buried under a deeper layer of skin and cause cysts.  These are usually treated by excision.  

I would suggest visiting with either a plastic surgeon, facial plastic surgeon, or ENT physician. If the lesion is truly a cyst, excision would be the best way to completely remove it. The lesion could be other things as well and an examination by a surgeon who works in the head and neck can help to develop the best option for removal.

You can see a dermatiologist, plastic and cosmetic surgeon or an ENT surgeon for possible excision of the cyst.  Cysts, by nature, have inner linnings that allow them to recur unless completely removed.