I now the logical name and Virtual IP of the cluster.
How can I find the active sun cluster node without having root access?
If you can at least log in to the cluster nodes you can try this command:
/usr/cluster/bin/clrg status
I'm not sure if you have to be root to do this or not. I have never tried to do anything cluster related other than as root. You could also look at each nodes process tree with a good old fashioned ps -ef and see if the service is running or not. If the application is set up as a scalable application, it will be running on several nodes (should be anyway).
See if this works,
You have the IP of logical hostname? now do 'ifconfig -a' on both nodes of cluster and find which node the IP is currently configured on or paste the output of ifconfig -a here and IP of logical hostname as well 
# ssh [logical name] hostname