Google indexes and Updates Pages that are 301 > 200 Status
If you have 2 domains such as mysite.cloudapp.net and mysite.com using 301 from the cloudapp to the mysite.com will be updated by Google because the Google's bot follows the redirects and as soon as it read a status 200 that becomes the master page and that's what will get indexed/updated.
So while a simple 301 will do the trick its not necessary the best method, I'm a huge fan of preventing duplicate content ending up in the index to begin with. I use htaccess or any other rewrite method to correct problems later on the line and not at the start with as its just not necessary at all while many people choose not to allow people visit the website via other means other than intended if they don't end up in the index then they don't get used in any case.
You should opt to use Rel Canonical within the head of your pages this way Google finds the master even if you the page may be accessible via many ways. If then Google detects a duplicate page only the master page will be indexed. If you want to enforce that domain being used then use a redirect as well as Rel Canonical.
How to Update Google
I thought I'd mention this because we get hundreds of posts asking how to make Google update search results and how long it takes... Basically if your site is new then Login to Google Webmaster Tools and you can use the URL remove method which can be found within Optimization > Remove URLS. You will need to add the cloudapp domain to webmaster tools to do so. Alternatively just use a 301 and Rel Canonical and just wait for Google to update it. It's hard to say if removing the urls is faster or just leaving it as the Google bot tends to come and go as it pleases and its hard to predict on new sites. It's worth mentioning that no one on this site can tell you how long it will take for this process to correct itself but normally its within 2 weeks but some people get angry and has been for a month or even more... Just be patient and work on building your new site further.