Finding the exact numbers on this will probably be unlikely without access to their analytics, but you can get a guesstimate by seeing which of their pages are used as site links when you search for their brand in Google. You can also derive some guesses via search rankings by dividing out a percentage of clicks based on ranking from total search volumes.
Still, you're going to be missing a lot of data by not knowing how well they're doing in terms of email marketing, social, referrers, etc. Companies that try to accomplish this are ones like Compete and Quantcast. A better way to go about finding popularity by topic though would be to use a tool like Google Trends, Twitter Trending, etc.
As for time on page, why? You'll be better off focusing all your efforts there on your own site's usability, experience and customer satisfaction.