Is there any solution (program/Docker image) that will take a port, forward it to another host (or maybe another program listening on the host) that then modifies the traffic to contain the real source IP. The whole idea is that in the server logs I want to see people’s real IP addresses, not the server in the cloud private VPN IP.

Not that I'm aware of. Most methods require some kind of out-of-band way to send the client's real IP to the server. e.g. X-Forwarded-For headers, Proxy Protocol, etc.

If your backend app supports proxy protocol, you may be able to use HAProxy in front on the VPS and use proxy protocol from there to the backend. Nginx may also support this for streams (I don't recall if it does or not since I mainly use HAProxy for that).

Barring that, there is one more way, but it's less clean.

You can use iptables on the VPS to do a prerouting DNAT port forward. The only catch to this is that the VPN endpoint that hosts the service must have its default gateway set to the VPN IP of the VPS, and you have to have a MASQUERADE rule so traffic from the VPN can route out of the VPS. I run two services in this configuration, and it works well.

iptables -t nat -A PREROUTING -d {VPS_PUBLIC_IP}/32 -p tcp -m tcp --dport {PORT} -j DNAT --to-destination {VPN_CLIENT_ADDRESS}
iptables -t nat -A POSTROUTING -s {VPN_SUBNET}/24 -o eth0 -j MASQUERADE

Where eth0 is the internet-facing interface of your VPS.

Edit: One more catch to the port forward method. This forward happens before the traffic hits your firewall chain on the VPS, so you'd need to implement any firewalls on the backend server.