Can you get rid of a sinus infection without medicine?

Depends. If it is triggered by a virus it will self heal. If bacterial, it might self heal over an extended period (months) like it must have before antibiotics were available. Yet some bacterial infections could progress over days and weeks to worrisome complications.

Yes, for the majority of them. It gets complicated by allergies sometimes that can mimic symptoms of an infectious sinusitis. If symptoms persist more than 2 weeks, seek medical attention.

Viral sinusitis will often go away in a week or so. Bacterial sinusitis can also resolve without antibiotics especially if you can treat the nasal-sinus congestion quickly. The congestion/swelling narrows the sinus openings and mucus plugging up these openings = risk for infections. So treat congestion/mucus early on and likely you'll prevent/treat the infection. Good luck.