How is your network setup that the server only has a public IP address? That implies that you aren't using NAT, or the server is in a DMZ without NAT. In either case, the network should be configured to allow access from private IP addresses to the DMZ public IP addresses.

One common solution is to use a split-horizon DNS. The internal DNS returns private IP address, and has record for purely internal hosts. The public DNS reutrns public IP addresses.

Another solution for some protocols is to use a proxy. The proxy would be setup outside (or on the firewall) and connect back inside. This only works for some protocols (HTTP), and raises security issues about securing the proxy and the services from untrusted outside usage.