How long does it take for arthritis to cure?

Depends. Arthritis is a very broad term that includes many different kinds of arthritis ( osteoarthritis, gout, psoriatic, rheumatoid, traumatic, viral, gonorrheal, etc.). Some forms never cure while others are transient. As you are young it really could be anything. Need a diagnosis in order to be able to predict the outcome.
Varies. For some it can be a chronic condition, for others it can be put in remission.