You were trying very correctly - You indeed need to modify the $_['text_success']
in mentioned language file, so that it looks like this (re-using Your code):
$_['text_success'] = 'Success: You have added <a href="%s">%s</a> to your <a href="%s">shopping cart</a>!<div class="checkout"><a class="button" id="close_cart">Continue Shopping</a><a href="index.php?route=checkout/checkout" id="checkout-button" class="button">Buy Now</a><a href="index.php?route=checkout/cart" class="button">View Cart</a></div>';
notice that all the <?php echo ... ?>
are replaced by corresponding strings only.
For closing this success notice by clicking on Continue Shopping link You need to register a click event listener, e.g. in catalog/view/javascript/common.js
add this somewhere at the end:
$(document).ready(function() {
$('#close_cart').on('click', function() {
$('.success').remove();
});
});
This was not the scope of the question so consider it as extra value to the answer...