Can you treat a collapsed lung without hospitalization?

Technically maybe. Up to 10+% of spontaneous pneumothoraces may be completely without symptoms. It is believed many patients with mild or no symptoms may remain at home and not seek medical assistance. Patients with small, asymptomatic pneumothorax, may be sent home after short evaluation and repeat x-ray in er and no other interventions.
Yes. You need to see a doctor who can explain this to you and monitor your treatment. You do not have to be in a hospital for many treatments.
If small sealed leak. A partially collapsed lung is due to an air leak from the inside of the lung through the covering of the lung, into the space between the lung and the ribs. The site of the leak has some damage, which will heal itself. If a very small amount of air leaked into the chest cavity, that air might go away without treatment. Depending on the case, a doctor might send a patient home from the emerg. Room.