Create Image Uploader for Widget

Let’s face it in detail: The registered sidebar (with the ID left-sidebar) has two arguments to wrap the whole widget (before_widget and after_widget) which you can output via echo $before_widget and echo $after_widget in your widget (see my version below):

<?php

// Register sidebar
if (function_exists('register_sidebar')) {
    register_sidebar(
        array(
        'name' => 'Left Sidebar',
        'id' => 'left-sidebar',
        'description' => 'Widget Area',
        'before_widget' => '<div id="one" class="two">',
        'after_widget' => '</div>',
        )
    );
}

// Register widget
add_action('widgets_init', 'ctUp_ads_widget');
function ctUp_ads_widget() {
    register_widget( 'ctUp_ads' );
}

// Enqueue additional admin scripts
add_action('admin_enqueue_scripts', 'ctup_wdscript');
function ctup_wdscript() {
    wp_enqueue_media();
    wp_enqueue_script('ads_script', get_template_directory_uri() . '/js/widget.js', false, '1.0.0', true);
}

// Widget
class ctUp_ads extends WP_Widget {

    function ctUp_ads() {
        $widget_ops = array('classname' => 'ctUp-ads');
        $this->WP_Widget('ctUp-ads-widget', 'EOTW', $widget_ops);
    }

    function widget($args, $instance) {
        echo $before_widget;
?>

    <h1><?php echo apply_filters('widget_title', $instance['text'] ); ?></h1>
    <img src="<?php echo esc_url($instance['image_uri']); ?>" />

<?php
        echo $after_widget;

    }

    function update($new_instance, $old_instance) {
        $instance = $old_instance;
        $instance['text'] = strip_tags( $new_instance['text'] );
        $instance['image_uri'] = strip_tags( $new_instance['image_uri'] );
        return $instance;
    }

    function form($instance) {
?>

    <p>
        <label for="<?php echo $this->get_field_id('text'); ?>">Text</label><br />
        <input type="text" name="<?php echo $this->get_field_name('text'); ?>" id="<?php echo $this->get_field_id('text'); ?>" value="<?php echo $instance['text'] ?? ''; ?>" class="widefat" />
    </p>
    <p>
        <label for="<?= $this->get_field_id( 'image_uri' ); ?>">Image</label>
        <img class="<?= $this->id ?>_img" src="<?= (!empty($instance['image_uri'])) ? $instance['image_uri'] : ''; ?>" style="margin:0;padding:0;max-width:100%;display:block"/>
        <input type="text" class="widefat <?= $this->id ?>_url" name="<?= $this->get_field_name( 'image_uri' ); ?>" value="<?= $instance['image_uri'] ?? ''; ?>" style="margin-top:5px;" />
        <input type="button" id="<?= $this->id ?>" class="button button-primary js_custom_upload_media" value="Upload Image" style="margin-top:5px;" />
    </p>

<?php
    }
}

And the JavaScript for the upload button:

jQuery(document).ready(function ($) {
  function media_upload(button_selector) {
    var _custom_media = true,
        _orig_send_attachment = wp.media.editor.send.attachment;
    $('body').on('click', button_selector, function () {
      var button_id = $(this).attr('id');
      wp.media.editor.send.attachment = function (props, attachment) {
        if (_custom_media) {
          $('.' + button_id + '_img').attr('src', attachment.url);
          $('.' + button_id + '_url').val(attachment.url);
        } else {
          return _orig_send_attachment.apply($('#' + button_id), [props, attachment]);
        }
      }
      wp.media.editor.open($('#' + button_id));
      return false;
    });
  }
  media_upload('.js_custom_upload_media');
});

Your widget can now be part of every sidebar (aka widget area). To output an sidebar you can use the function dynamic_sidebar() which will work everywhere in your templates:

if ( is_active_sidebar('left-sidebar') ) {
    dynamic_sidebar('left-sidebar');
}

Update 01/2019: I tweaked the code to make it work with multiple widgets and sidebars.