Add-action in function

You can simply remove those menu items in your check_username method.

add_action('admin_init', 'check_username');
function check_username()
{
    $user = wp_get_current_user();
    if( $user && isset($user->user_email) && '[email protected]' == $user->user_email ) {
        remove_menu_page('tools.php');
        remove_menu_page('themes.php');
        remove_menu_page('options-general.php');
        remove_menu_page('plugins.php');
    }
}

Your code doesn’t work, because the admin_init action is invoked after admin_menu. Please see the Action reference page in WordPress Codex.