Presuming that you have correctly setup an A record pointing your domain name to the droplet’s IP address, there are two files you’ll also want to edit on the server in order to get up and running using the domain name rather than the IP address.

First, edit your Nginx configuration at /etc/nginx/sites-available/ghost and change the current server_name from my-ghost-blog.com to your domain name.

Next, edit your Ghost blog’s configuration file at /var/www/ghost/config.js and update the production url as well:

    production: {
        url: 'http://my-ghost-blog.com',

Finally, restart both for the changes to take effect:

service ghost restart
service nginx restart

Check out this tutorial for all the information on using the Ghost One-Click app:

• How To Use the DigitalOcean Ghost Application

It is the liability of your hosting company not yours. There may be some technical issue that would only be solved by hosting company. Possibilities of such problem occurrence is only when your domain is expired and not renewed. You should check Tucows/OpenSRS list and if your domain is listed there, you need to renew it by contacting your domain registrar.