Welcome to WebmasterWorld Guest from 18.104.22.168
Today I was forwarded with Google's Webmaster tool alert message saying that the Googlebot couldn't access the site
This part, by itself, can be ignored. It's a long-standing known issue. You can test it for yourself using Fetch As Googlebot.
You have to put in both versions explicitly, even though g assumes it is the same domain
it's definitely worth verifying both www and non-www hostnames to detect any issues with that
Yes, you yourself have to maintain Duplicate Content in wmt ;)
Some areas, notably Links To Your Site, carry separate information for the two forms of the name.