First make sure your DNS has an * record for www pointing at your server IP, this will make sure your site still displays without a www.
If you wanted to add the www bit on for consistency of the URL then look at using .htaccess if it's applicable to your situation.
The quick fix (entirely under your control) would be to 'fix' your local settings so http://mysite works.
What web-server are you using?
You might also ask this question over in serverfault.com .. unless it get's moved for you.
You should make sure
1) that you accept both the address with www and without.
2) then pick the one that is the right one - the one you want
3) then you 301 redirect the wrong one to the right one.
That way it can be accessed from both urls, but you tell both people and search engines which one is the right one, by saying that the wrong one is permanently redirected to the right one.
Secondly, you avoid duplicate content, which can harm your search engine ranking.
In addition to setting up the rewrite rule to redirect traffic to www.example.com. I would also have a look at the Bing webmaster tools. This should allow you to make sure that Bing knows about your website and lets you specify the url for it's links. It also provides you information such as crawling errors, lists where your site is linking too and where it's being linked from. Among other things.
