You can do this on background using some ajax on page load, here is an untested example to show the idea.
jQuery(function() {
jQuery.get('http://domain.tld/remote');
});
<?php
function is_ajax()
{
return (isset($_SERVER['HTTP_X_REQUESTED_WITH']) and strtolower($_SERVER['HTTP_X_REQUESTED_WITH']) == 'xmlhttprequest');
}
function add_rewrite_rules($wp_rewrite)
{
$new_rules = array
(
'remote' => 'index.php?remote=true'
);
$wp_rewrite->rules = $new_rules + $wp_rewrite->rules;
}
function add_query_vars($vars)
{
array_push($vars, 'remote');
return $vars;
}
function remote_feed()
{
if (($var = get_query_var('remote')) and is_ajax())
{
// get remote feed and insert/update posts
}
}
add_action('generate_rewrite_rules', 'add_rewrite_rules');
add_filter('query_vars','add_query_vars');
add_action('template_redirect', 'remote_feed');