Can a drupal admin user get FTP rights to a godaddy server
I don't use Drush. But using the install module function at admin/modules/install I have installed modules before when live on a GoDaddy server.

I don't know what is causing the problem, I would maybe contact their support. But certainly it's doable, and I've done it in the past.

However depending on the module I would stay clear of installing on a live server, particularly if you are on shared hosting environment with GoDaddy (contrary to a lot of people I actually find them okay and the best solution at that price for hosting a site).

However whilst you'll find installing modules okay you may find you run into problems enabling them with PHP memory limits, or other server issues. Generally doing dev on a local machine if you have hosting like GoDaddy is going to be faster, more flexible, and generally easier.