In order to do this, you’d need to install some kind of plugin that allows you to modify the Capabilities that each role has access to. There are many plugins that do this, but my personal favorite is the Members plugin.
After installing this, you’d need to modify the ‘Contributor’ role to remove the ‘Edit Post’ capability (http://d.pr/i/hqYy).
Then assign a user to the ‘Contributor’ role and when they login they should only see ‘Events’ in the menu, like this: http://d.pr/i/LMmj
Hope that helps! Just let me know if you have further questions.