Yes, but you'll need a Personal Account, too.
Facebook offers two kinds of accounts: one for individuals, called "Personal Accounts," and the other, for businesses for the purpose of administering a Page, called "Business Accounts."
SharePress relies on a feature of Facebook called Facebook Applications or "Apps." In order to use SharePress, you need to create your own Facebook Application. And in order to create a Facebook Application, you need to have a Personal Account.
So, if you only have a Business Account, you'll need to follow these extra steps for setting up SharePress.
Log out of Facebook, and sign up for a Personal Facebook Account.
Log out of Facebook again, and login as your Business Account.
Navigate to your Page's settings, and make your new Personal Account an administrator of your page.
Log out of Facebook again, and login as your Personal Account.
Accept the invitation to be an administrator of your Page.
When that's all done, you can continue on with and complete the remainder of SharePress' normal setup.
As an administrator of your Business Account's Page, your Personal Account now has the ability to grant permission to your Facebook Application to post things to your page. This permission is at the heart of how SharePress works.
When you run SharePress' setup and authorize your Facebook Application, you are granting your application permission to (1) see a list of the pages your Personal Account can administrate, and (2) post to those pages on your behalf.
You can be sure everything is set after SharePress' setup has completed. In WordPress, go to Settings, SharePress, and scroll down to Facebook Pages and Walls. There you will find a list of all the destinations to which SharePress may post your content. Among them will be your Business Account's page.
This is what this looks like in my personal blog: