The second link (Service) should be changed to ‘Ästhetik’ and should have a new href.
Here’s what i got:
<script>
var cont = document.getElementsByClassName('breadcrumb');
var el = cont.getElementsByClassName('ul')[1];
var link = el.getElementsByTagName('a');
link.innerHTML = "Ästhetik";
link.href = "http://bo-institut.de/aesthetik/";
</script>
cont is never defined because you changed the name.
ul and a aren’t class names so getElementByClassName won’t help you, getElementsByTagName will do what you want.
Combining them all into the single var declaration will not allow you to get the value of each so separate each onto their own lines.
indexes in JavaScript start at zero so I think you’re off by one on each.
var breadcrumb = document.getElementsByClassName('breadcrumb')[0];
var ul = breadcrumb.getElementsByTagName('ul')[0];
var a = ul.getElementsByTagName('a')[1];
a.innerHTML = "Ästhetik";
a.href = "http://bo-institut.de/aesthetik/";
Can be simplified using querySelector / querySelectorAll
var a = document.querySelectorAll('.breadcrumb a')[1];
a.innerHTML = "Ästhetik";
a.href = "http://bo-institut.de/aesthetik/";