If you choose Manually from the Sort Notification Center pop-up menu, you can do exactly that. Within the preference pane, click and drag applications so that they appear in the order you prefer. If you want Calendar notifications to always appear at the top of Notification Center, for example, just drag Calendar to the top of the list. (And yes, it can appear above the Share Buttons entry, but it won't appear above the hidden 'Show Alerts and Banners' switch.)
There you have the ins and outs of Mountain Lion's notifications--a handy way to keep up-to-date on the events and messages that are most important to you.
Sign up for CIO Asia eNewsletters.