How fast can I make my pneumothorax go away?

Rest! A lung can take weeks to fully recover from a pneumothorax, depending on the cause, extent, and treatment of it. The primary goal is to 'reinflate' the lung and prevent air from again leaking into the pleural space. This will require rest, and regardless of the severity, caution: having had a pneumothorax increases the likelihood of developing another later in life.
Oxygen, rest. Wearing high flow oxygen increases the rate of reabsorption of the pneumothorax. The most common gas in the atmosphere is nitrogen, and if you change the nitrogen partial pressure in the blood and lung the gas will flow into the blood more quickly. Exercise with a pneumothorax may make it worse, delaying recovery. No flying or diving until you are cleared by a lung doc.