this works for me

http://codepen.io/ryanjbonnell/pen/dqsxI

jQuery('.gallery-link').on('click', function () {
    jQuery(this).next().magnificPopup('open');
});

jQuery('.gallery').each(function () {
    jQuery(this).magnificPopup({
        delegate: 'div',
        type: 'image',
        gallery: {
            enabled: true,
            tPrev: '',
            tNext: '',
            tCounter: '%curr% | %total%'
        }
    });
});

Probably too late to initial asker, but might help someone else...

No href attribute to button, you need to use "mfp-X" class names instead.

For me "mfp-inline" did the trick, but for ajax you need probably something like this:

<button class="ajax-popup-link mfp-ajax green" data-mfp-src="#div_element">Enter Now</button>
...
$('.ajax-popup-link').magnificPopup();

(Not sure if in ajax you need this, but there is also "data-mfp-src" attr that shows where dialog div is...)

If you're using jQuery you could just listen for the window load event and then call the open method for your Magnific Popup like so:

(function($) {
    $(window).load(function () {
        // retrieved this line of code from http://dimsemenov.com/plugins/magnific-popup/documentation.html#api
        $.magnificPopup.open({
            items: {
                src: 'someimage.jpg'
            },
            type: 'image'

          // You may add options here, they're exactly the same as for $.fn.magnificPopup call
          // Note that some settings that rely on click event (like disableOn or midClick) will not work here
        }, 0);
    });
})(jQuery);

When you use an array of items you can pass the index of the first item you want to show. So I have used var index = jQuery(this).parent().index() to get the index of the current clicked item and then passed that variable in to the magnificPopup function.

To get the caption in the popup I have added an extra property to the items object called titleSrc, which you can then retreive in the titleSrc option using item.data.titleSrc.

https://jsfiddle.net/sjp7j1zx/4/

jQuery(".caption a").on("click", function(event) {
  event.stopPropagation();
});

jQuery(".caption").on("click", function(event) {
  var items = [];
  jQuery(".item").each(function() {
    // Pass an extra titleSrc property to the item object so we can use it in the magnificPopup function
    items.push( {
      src: jQuery(this).find("a").first().attr("href"),
      titleSrc: jQuery(this).find('.caption').html()
    } );
  });

  // Get the index of the current selected item
  var index = jQuery(this).parent().index();

  jQuery.magnificPopup.open({
    type: 'image',
    gallery: {
      enabled: true
    },
    items: items,
    image: {
      titleSrc: function(item) {
       // get the titleSrc from the data property of the item object that we defined in the .each loop
       return item.data.titleSrc;
      }
    }
    // Pass the current items index here to define which item in the array to show first
  }, index);
});

You can use the focus option of magnific-popup.

$.magnificPopup.open({
    items : {
    type : 'inline',
    src : '#idOfInlinePopUP'
    },
    focus: '#closeButton',                                                                   
    closeOnBgClick:false,                                                                        
    enableEscapeKey:false
    }, 0);

Code of button:

<input type="button" value="close" onclick="$.magnificPopup.close();" id="closeButton">

In this code the focus option consists of the ID of the button which you want to open. And src has to contain the id of the inline popup.

If you are able to open your popup successfully then you just need to add focus attribute with ID of the button you want to click on enter click.

In this answer, on opening of popup the default focus will be on closeButton. On clicking this button the popup will close.