Quit Aperture - force quit if necessary .
Launch Aperture again.
When the "Mobilee Me" panel appears, press the "Learn more" button, not o.k.
Then use the "File > Web Accounts > Manage Web Accounts" menu entry from Aperture's main menu bar and remove "MobileMe" from the list of accounts in the accounts pane by pressing "-".
Afterwards sign out of Mobile Me from the "System Preferences > Mobile Me" panel.
If this does not work, disconnect the internet (Wi-fi, Ethernet, whatever you have), before you launch Aperture. Now the panel should no longer appear.
P.S. Instead of Force Quitting
you might try this:
If Aperture is unresponsive, how to avoid force quitting?