She can, and according to the link I'm about to post, he can stay until her sponsorship of him has been determined. Once the sponsorship is granted he gets to stay even if he divorces her during the three year sponsorship, and she's still obligated to support him. She's gullible enough to believe he truly is in love with her as well.
This link refers to woman marrying in Canada on a temporary resident (aka tourist) visa. I can't see that it would be different if the tourist is the male and the Canadian is the female in the relationship.
If you decide to meet your husband-to-be in Canada before making your final decision, you will most likely come to Canada on a visitor visa, known as a Temporary Resident Visa. You may not have to have a visa if residents from your home country are not required to have a visa upon entering Canada.
www.lawforforeignbrides.ca
If this is the case then I can see why the rush is on to get her financial issues settled. As soon as she is in a position to sponsor him, he's got his ticket to Canadian permanent residency.