For a smooth slide-up:-
$("#success-alert").fadeTo(2000, 500).slideUp(500, function(){
$("#success-alert").slideUp(500);
});
$(document).ready(function() {
$("#success-alert").hide();
$("#myWish").click(function showAlert() {
$("#success-alert").fadeTo(2000, 500).slideUp(500, function() {
$("#success-alert").slideUp(500);
});
});
});
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.3.1/css/bootstrap.min.css" integrity="sha384-ggOyR0iXCbMQv3Xipma34MD+dH/1fQ784/j6cY/iJTQUOhcWr7x9JvoRxT2MZw1T" crossorigin="anonymous">
<div class="product-options">
<a id="myWish" href="javascript:;" class="btn btn-mini">Add to Wishlist </a>
<a href="" class="btn btn-mini"> Purchase </a>
</div>
<div class="alert alert-success" id="success-alert">
<button type="button" class="close" data-dismiss="alert">x</button>
<strong>Success! </strong> Product have added to your wishlist.
</div>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://stackpath.bootstrapcdn.com/bootstrap/4.3.1/js/bootstrap.min.js" integrity="sha384-JjSmVgyd0p3pXB1rRibZUAYoIIy6OrQ6VrjIEaFf/nJGzIxFDsf4x0xIM+B07jRM" crossorigin="anonymous"></script>