What is a "the ponytail lift"? What are typical costs for it? (photos)

I have never heard of a pony tail lift and after researching it, it appears to be a marketing gimmick to somehow set this doctor aside from others as doing something "different".  The description of what it is, is what many doctors do all the time.  To create a more upward slanted eye you would require a lateral canthopexy.  Your plastic surgeon can describe the procedure to you in detail so you can decide if this would be what you wish.

A
ponytail lift is not a medical term, and to my knowledge, I don’t know of any
surgeons who call their signature lift this name. upturned eyes, firmer jawline
and reduction in facial expression lines can be achieved when hair is put in a
tight ponytail, and in turn, these can also be achieved with surgical facelift
techniques.

Do
update your question with photographs so that we can further expand on this
query before you book a consultation to see a surgeon.

I have not heard of a Ponytail lift - this sounds like a branded or trademarked name for some form of face or eye lifting procedure. What you describe you are looking for would be called a lateral canthopexy - this is a procedure to reset the outside corner of the eye and to provide support to the lower eyelid.  In performing this procedure the position of the corner of the eye can be slightly raised or lowered to give the eye a more upward slant.