Does outbound port have to be open to make an Json-RPC call
The RPC server doesn't care what source port the request comes from, and it doesn't need to make any extra connections back to the client on that port or any other.

However, a possible cause for your trouble is that the shared webserver's firewall blocks outgoing traffic with a destination port that is not on their whitelist.  (That may be what is meant by "having specific outbound ports open".)

The fix would be to change the RPC server configuration to listen on some other port which is allowed  as a destination port by the shared webserver.  The most obvious choice would be port 80, if the RPC server machine isn't also running an HTTP server.  Otherwise, you can try various commonly used ports to find one that is allowed by the firewall and not already in use.