The cloud dashboard is publicly available on the Internet and can be reached at: https://dashboard.cloud.catalyst.net.nz
The cloud APIs are available to all compute instances hosted on the cloud, but need to be white-listed to your IP address for access over the Internet. If you have provided an IP address during sign up, you should be able to reach the APIs from this IP. Otherwise, you can open a support request via the dashboard at any time to request a change to the white-listed IPs.
The compute instances you launch on the Catalyst Cloud are created in your private network by default. You have the option to associate a floating IP (public IP) to expose them to the Internet. You can use security groups (similar to firewalls) to define who has access to your compute instances, as explained in Security groups.