Before I start, I always remind people that cancer is NOT one specific illness with one specific cause or treatment. In fact, most people would be amazed at how many types of cancer there are.
One of the first depictions of cancer in historical medical literature dates back to about 1600 BC, from what's known as the Edwin Smith Papyrus. For a historical overview, the Journal of Surgical Oncology has a paper here.
The name "cancer" was attached to various malignancies by Hippocrates, from whom we have documented cases from ancient Greece. So, the first answer is simple: No, cancer is not a modern, man-made disease.
As for the quote posted, I would like to point out a simple flaw in what seems to be the central statement of this argument: "There is nothing in the natural environment that can cause cancer". This is easily falsifiable. I think most rational people would agree that there are few things more natural than SUNLIGHT. Of course, there is a fairly well-documented association between sun exposure and melanoma. There are also known viral causes of cancer, the most well known being HPV. While we may not like to think of a virus as a pleasant thing, it is in fact perfectly all-natural.
Any sufficient analysis of cancer rates is going to be much too complex for a simple post, but I can suggest some places to begin research, if one is so inclined.
For some data as to current trends; you can find a table of death rates for males and females between 1930-2006. There is also a nicely put together table here for cancer probabilities for 2004-2006 broken down by age, sex, and organ.
The NCI offers a searchable database of available statistics from the SEER which should also provide a good place to start looking if one is curious.
Remember that cancer is a hot-button issue in the media, and what is reported in the general press may not accurately reflect the current state of the peer-reviewed legitimate research. Search out the citations if you can and see for yourself.
Also keep in mind that studying prevalence and incidence rates for cancer is very time-consuming and labor-intensive. One must take into consideration many factors in order to properly interpret the data. Some confounding factors are:
Increased diagnostic capabilities which allow modern medicine to find and diagnose more cancers.
Increased lifespan which allows for more time to develop cancer
Immunosuppressive conditions such as HIV which make patients more susceptible to cancers.
Better survival rates for cancers which increase the number of cases in the general population.
This is obviously not a complete answer, but I hope it was helpful as a place to begin.