Well, Pandora is a public company, so they have to disclose all that stuff. Which means we can look it up! Transparency is awesome.
According to their latest annual report (http://phx.corporate-ir.net/External.File?item=UGFyZW50SUQ9M...), in fiscal 2012 Pandora brought in $274 million on $285 million in operational expenses.
Of those expenses, $149 million were for content licensing. Product development accounted for $13 million.
Of the money they made, the vast majority ($240 million) came from advertising.
Edit: Also, these numbers aren't just cash. Revenue is realized when it's earned, not when it's actually paid. So, for example, if Pandora provides $50MM of advertising, and then bills that, it records $50MM of revenue, even if it hasn't received all of that yet. Instead of being counted in cash, it's counted in accounts receivable (commonly abbreviated as A/R). This adheres to GAAP (in the US).
If you truly want to support the artist, go to every single one of their concerts and buy three t-shirts.
I'm guessing I've bought dozens of CDs and gone to multiple concerts thanks to Pandora.