Obviously it's impossible to specifically say. There are very few cases where we can retroactively review medical cases for any sort of evidence to diagnose cancer and get a realistic viewpoint on how "big of a deal it was". Furthermore, the diagnostic methods and understanding that modern medicine now possesses make any meaningful comparison very difficult. They simply never could have found some forms of cancer we know about today.

Hippocrates used the term carcinos in his Corpus texts and spoke about a black bile that he attributed to the masses. Hippocrates other writings seem to clearly present that Ancient Greeks were aware that removing these tumors was possible and would benefit the health of the patient, describing what we could infer is a Mastectomy. He even had the knowledge to clearly outline when he thought a tumor should or should not be removed, and he was able to identify several "superficial" and "deep-seated" types of tumors of the vagina, breasts, stomach, armpit, groin, and other soft tissue.

The Greeks and Romans prescribed a variety of potions, salves, concoctions, rites, and other things to try to mitigate cancer, but none of them appeared to be even remotely successful. I don't have any specific references to these curatives/treatments in any meaningful fashion. Hippocrates did advocate, in the event medicine didnt work, treating tumors via knife. And if that did not work, cautery.

I can only speculate on the state of mind the ancient Greeks or Romans would have had about cancer, but from the choice of phrasing, it did not appear particularly optimistic. "No treatment" is a commonly used term.

I'm tangentially aware of other cultures with older 'knowledge' of cancer, especially Egypt, but I don't know that material well enough to comment and would absolutely need to call someone else in on that.

I recommend first reading Hippocrates' Corpus (numerous translations are available, all mostly okay) and then perhaps following it up with Celsus's work, De Medicina.