Is there cartilage or bone sticking through my left nostril?

Cartilage that can be seen when one looks up the nostril is often a subluxed septum. However, what you describe with pain and the fact is comes and goes, does not sound like a cartilaginous structure--it may be a cyst or folliculitis. I would recommend the next time it flares up to see an otolaryngologist or facial plastic surgeon.

It would be impossible to offer you a meaningful advice without examining your nose. I suggest that you consult one of us locally to allow us to see what is causing the problem and give you a recommendation. This could be an infected small cyst around the hair follicle or an abnormal piece of cartilage causing a small erosion of the lining of the nose. It does not seem like something that you need to worry about but it is definitely something should be looked into.

Without  a picture of the problem, it is impossible to say what is causing this. Trying to make a diagnosis only based on online opinions is not a good idea. I suggest making an appointment with a plastic surgeon or facial plastic surgeon.