Take a day to shake it off. Getting back a poor grade can certainly provoke a negative reaction, but you want to dispute a grade as calmly and rationally as possible.
Schedule an appointment with your professor during her office hours to discuss your grade. Make sure this is done in person so you can have a rational discussion about it. E-mail debates can quickly turn sour.
Write up a list of reasons why you think you deserve a better grade before meeting with the professor. To do this, take the list of requirements for the paper and check off reasons why you think it fulfilled each one. If you failed to acknowledge certain requirements, bring this up as well. It is better to be honest and recognize your failings than skirt the issue. You may even gain more respect for doing so.
Practice discussing these points in the same way you would practice for a speech or debate. Consider each counter-point your professor may bring up and consider your own thoughts on those points.
Meet with you professor and explain your points in a rational, calm manner. While you should be assertive, don't be accusatory or impatient. Discuss why you deserve a better grade and address her reasoning for giving you a lower grade.
Offer to re-edit or add to the paper if the professor still doesn't see your point of view. That doesn't mean admitting failure; it simply means you both have different opinions on what the requirements meant. Especially if this is the first paper of the course, a professor may offer some leniency. After all, she or he has to adjust to writing styles as much as you do.
Don't push the issue. While you should stand up for yourself, remember not to become forceful in your opinions. After all, the professor is entitled to an opinion as well. If you maintain respect for each other, you'll have a better relationship. If there is no give on this issue, ask for help when drafting the next paper.