Please help us understand and confirm if my understanding is correct:
- you've deployed the Joomla stack using Bitnami over Azure
- it created a VM, which is linked through bitnami-joomla-123456789.cloudapp.net
- you would like the link to be different (e.g. joomla.cloudapp.net)
If i understood you correctly, there is not much you can do regarding the VM name. At least for Google Compute, you can't rename it. Thing is - you don't need to. My suggestion is for you to use a domain, pointing towards your server. In order to set up a domain for your Joomla Instance, please read this guide.
Again, if i understood correctly, you've already managed to set up your joomla instance into the root url (you access your instance through bitnami-joomla-123456789.cloudapp.net, not bitnami-joomla-123456789.cloudapp.net/joomla). This should've been the hardest part. If you haven't done it yet, it's in the above-mentioned guide. If you have, all that's left is to point a domain towards your server and you should be set.
If you don't know how to do it, i'm suggesting one universal solution (since i'm not familiar with Azure specifics): you can use Cloudflare DNS. Basically you buy a domain, add it to cloudflare and edit DNS records there. You need one A record and one CNAME record.
Shooting in the dark, but hope some of it may help you.