This looks like a great opportunity. The product management looks very interesting, but I think you might be looking for more experience. Anyway to meld some of those responsibilities with a business intern position for a very hungry new grad with a take charge attitude?
As a business person, the number one thing I think is realizing that to a programmer for the most part business is EVERYTHING other than coding, which includes all the stuff I put above, and it's that stuff that isn't taught in most business schools but what startups want.