Update Custom post field by Ajax on cached site

Question

I’ve spotted some errors here:

jQuery.ajax({
     type: "POST",
     contentType: "application/json; charset=utf-8", // default: 'application/x-www-form-urlencoded; charset=UTF-8'. you can not set
     url: "http://localhost/wp-admin/admin-ajax.php", // if you have correctly enabled ajax in wp, you should use the object you set up with the url
     data: "{'action':'get_PostViews(" + idpost + ")'}", // you can use a PlainObject notation, so you don't need to double quoted. action property is the name of your function as you written in function.php
     success: function (result) {
       alert('Update Success!');
     }
});

Take a look here jQuery.ajax() .
To use ajax within WordPress follow these steps:

enable ajax functionality
declare your function in function.php
use javascript/jquery to send data to server and to listen data
retrieved

Enabling ajax

The best way to accomplish that ( in my opinion ) is:

//File functions.php
    add_action('template_redirect', 'ajax_activation');
    function ajax_activation(){
       //optional 
       wp_enqueue_script( 
          'ajax_script', 
           get_template_directory_uri() . '/js/jquery.ajax.js', // path to your js file for ajax operations
           array( 'jquery' ), false
       );
       //end optional
       wp_localize_script( 
          'ajax_script', // the name of your global.js registered file
          'ajax_object', // name 
           array( 'ajaxurl' => admin_url( 'admin-ajax.php' ) ) // you can add other items for example for using a translated string in javascript/jquery context
       ); 
    }

Declare function

//File functions.php
add_action('wp_ajax_get_PostViews', 'get_PostViews');
add_action('wp_ajax_nopriv_get_PostViews', 'get_PostViews');

function get_PostViews() {
    $id = isset( $_POST['id'] ) ? $_POST['id'] : false;
    // your code here
    wp_die(); // | die(); you need this to avoid trailing zero
}

jQuery/Javascript

$.ajax({
    type: "POST",
    url: ajax_object.ajaxurl, // this is the object you defined in function.php
    data: {
       action: 'get_PostViews', // the name of your function
       id: // you can store it in html attribute for an easy access like: jQuery(element).attr('id');
    },
    success: function (result) {

    }
});

I guess you are using this function for all posts in a loop, you can call ajax once to do the work for all posts.
For example i want to retrieve with ajax the titles of my posts:

HTML

<html>
<!-- some stuff here -->
<h3 id="<?php echo get_the_ID(); ?>" class="spyhole"></h3> <!-- there are many of this :  ) -->
<!-- some stuff here -->
</html>

jQuery

ids = [];
items = $('.spyhole');
$.each( items, function( i, v){
    ids.push( $(v).attr( 'id' ) ); // each value is added to array
});
$.ajax({
    type: "POST",
    url: ajax_object.ajaxurl,
    data: {
      action: 'getMyTitleAjax',
      id: ids
    },
    success: function (result) {
      data = $.parseJSON( result ); // Takes a well-formed JSON string and returns the resulting JavaScript object.
      $.each( data, function( i, v ){
        $('.spyhole[id="' + i + '"]').html( v ); // print the title
      });
    }
});

PHP

// Enabling ajax - functions.php
add_action('template_redirect', 'ajax_activation');
function ajax_activation(){
   //optional 
   wp_enqueue_script( 
      'ajax_script', 
       get_template_directory_uri() . '/js/jquery.ajax.js', // path to your js file for ajax operations
       array( 'jquery' ), false
   );
   //end optional
   wp_localize_script( 
      'ajax_script', // the name of your global.js registered file
      'ajax_object', // name 
       array( 'ajaxurl' => admin_url( 'admin-ajax.php' ) ) // you can add other items for example for using a translated string in javascript/jquery context
   ); 
}

// Declare my function
add_action('wp_ajax_getMyTitleAjax', 'getMyTitleAjax', 3);
add_action('wp_ajax_nopriv_getMyTitleAjax', 'getMyTitleAjax', 3);
function getMyTitleAjax() {
    $ids = isset( $_POST['id'] ) ? $_POST['id'] : false; // check if there is something in global $_POST
    if( $ids && is_array( $ids ) ){
      foreach( $ids as $id ){
        $titles[$id] = get_the_title( $id ); 
      }
    }
    echo json_encode( $titles ); // prints the result
    wp_die(); // avoid trailing zero
}

Hope it helps, if something is not clear feel free to ask

Update

According to your question updates, change this:

function get_PostViews() {
$id = isset( $_POST['id'] ) ? $_POST['id'] : false;
$count_key = 'post_views_count'; 
$count = get_post_meta($post_ID, $count_key, true);
if( empty($count) ){ $count = 1; } else { $count++; }
update_post_meta($post_ID, $count_key, $count);
}

with this:

function get_PostViews() {
$id = isset( $_POST['id'] ) ? $_POST['id'] : false;
$count_key = 'post_views_count'; 
$count = get_post_meta($id, $count_key, true);
if( empty($count) ){ $count = 1; } else { $count++; }
update_post_meta($id, $count_key, $count);
}