This sounds like a great opportunity to network and to learn more about the firm.
If I were you, I would send a brief email introducing yourself and asking for a phone call. In the call, try to not ask any questions with obvious answers (everything you can find on the website) and focus on the personal experience of the partner. Ask for advice on the application process and follow-up with an email after the phone call. A couple of weeks later, follow-up again with details on how you have implemented the advice that you have received in the phone call. When you apply, send a quick email to the partner, thank him again for his help and hope that he will refer you internally :)