The file on the server is ’~/.ssh/authorizedkeys’; you put your idrsa.pub in this (typically a server wouldn’t even _have an idrsa.pub on it, at least not for a user account).

Console access from the DO web interface is a local terminal, so you would need to use local authentication. If you did not set a passphrase (if you preloaded your keys when creating it, there is no password set, password auth is disabled), then you would not be able to get in. There should be an option in the web interface to reset the root password; attempt this, then login via console using that password. Change that password, fix your settings, and you should be good to go.