I've got no facial hair and I'm 21. Can I get a proper beard and how much will it cost?

Hair transplant to the face works.  The success rate and the overall results vary from doctor to doctor like any surgery.  The cost can range from $3000 to $10,000+ depending on your goals. Facial hair is a genetic trait that you inherited from your mother or father. It is not generally related to hormones or a disease process. Some may try to use Rogaine but without great success. You may want to consult with a doctor for possible surgical options.

Facial hair transplants involve the moving of
hair from one part of the scalp to the beard region. Performing it requires the careful harvesting
of grafts from the scalp, then placing these usually one and two hair grafts
into individual recipient sites. These
are challenging delicate procedures but can be immensely satisfying if
performed correctly and artistically.

Rogaine will not help to grow your beard.  Your best bet is with a hair restoration procedure where hair is borrowed from the back of your head and replaced into the desired areas on your face.  The technique used depends on your hair transplant surgeon, your hairstyle, and the number of grafts needed.  Consult with a facial plastic surgeon, plastic surgeon, or dermatologist to discuss your options in person. I wish you well!

I do not think that you will find success with products and suggest a hair transplant procedure.  This is done commonly with both the FUE and the FUT techniques. 

I think you are in luck!  Having full head of hair especially in the areas that we call them permanent zone allows us to transplant your scalp hair to create a natural looking beard.  You need to consult with a good hair transplant who has done this procedure and is aware of all its complexities.

Many young men want to accelerate their beard growth and ask
about it. It takes age (genetic triggers which occur in family lines) so
if you want to find out when your beard will grow out, you need to find out
when your father or grandfather started to grow theirbeard. Hormones should have kicked in
nicely by the time a man reaches 18 but he must wait out his genetics
as hormone alone are not responsible for the onset of beard grown. Beard
transplants can be done when the young man is a bit older (over 25) if the
beard has not grown in yet

Hair transplant is very successful for facial hair , bit you need to wait you may still grow some hair. There is no study that shows rogaine helps but you can try the 2 percent for a few months and see if it helps.