Admin menu for client, each menu for each content type
Since you just need a few links, I would use the Shortcut module which is a Drupal 7 core module. The documentation on this is pretty good: https://drupal.org/documentation/modules/shortcut.

You can configure under user permissions whether the user can edit the menu themselves or not. You can have different shortcut menus for different users. 

Out of the box, you can only have 7 shortcuts on the shortcut bar at one time. There appears to be a patch for this, but I haven't tried it yet myself: https://drupal.org/node/682000#comment-6493066

However, to display the shortcut toolbar, it appears you have to expose the Admin toolbar to that user. You can make the admin toolbar pretty minimal by just giving the tool bar and create content permissions to the user.

If you really don't want to expose the admin toolbar to this user at all, then you might want to go with a menu block that only this user sees (set under block config>Roles). Your user would need to be given a unique role for this (which other users are not given). This assumes you have the Menu, Block and Menu Block modules installed.