How do I Import an RSS feed as WordPress posts without duplicates?

The RSS feed has all the items in a fixed order(latest to oldest). In this case you can save the date & time of the last post you created, as an option & when you read the feed again, you can check the time previously saved to know which of the posts in the feed are new & insert them, then update the time again

The functions you’re interested in are update_option, get_option & wp_insert_post. You can find the reference for each of them in the wordpress codex, just google it. The flow would go like this

// retrieve the previous date from database
$time = get_option('mylastfeeddate');

// include the code to read the xml file here &
// then the foreach loop as you did in the question
foreach($items as $item) {

    // if the date is < than the date we have in database, get out of the loop
    if( $item->pubDate < $time) break;

    // assign the values in the format of wp_insert_post()
    $out = array();

    // insert the post
    $post_id = wp_insert_post( $out );

}

// update the new date in database to the date of the first item in the loop
update_option( 'mylastfeeddate', $items[0]->pubDate );

UPDATE

For making the code execute after some fixed time frame use wordpress cron functions like this

if (!wp_next_scheduled('update_feed'))
    wp_schedule_event(current_time('timestamp'), 'hourly', 'update_feed');

add_action('update_feed', 'function_name');
function function_name() {
    // here goes all the code to read the feed
}

change hourly to the time you like here is the codex reference http://codex.wordpress.org/Function_Reference/wp_schedule_event

For adding custom time frames

add_filter('cron_schedules', 'new_cron_schedules');
function new_cron_schedules($schedules) {
    array_merge($schedules, array('two_hourly' => array( 'interval' => 7200, 'display' => __('Twice Hourly') )));
}

After this filter for eg. you can replace ‘hourly’ above to ‘two_hourly’