Apache/HTTPD tends to be around in most enterprises or if you're using Centos/etc at home. So, if you have that around, you can do a proxy very easily to add the necessary CORS headers.
I have a blog post on this here as I suffered with it quite a few times recently. But the important bit is just adding this to your /etc/httpd/conf/httpd.conf file and ensuring you are already doing "Listen 80":
<VirtualHost *:80>
<LocationMatch "/SomePath">
ProxyPass http://target-ip:8080/SomePath
Header add "Access-Control-Allow-Origin" "*"
</LocationMatch>
</VirtualHost>
This ensures that all requests to URLs under your-server-ip:80/SomePath route to http://target-ip:8080/SomePath (the API without CORS support) and that they return with the correct Access-Control-Allow-Origin header to allow them to work with your web-app.
Of course you can change the ports and target the whole server rather than SomePath if you like.