If you are using Outlook 2010 or later your quota and current mailbox size is displayed when you go to; File- There is no way to see your mailbox quota directly in Outlook 2007 or previous.
To resolve this, you will need to increase the amount of disk space for that email account to receive mail again.
You can now see the amount of free space left and the number of items you have in each folder, in the bottom left hand corner.
Your mailbox contains all the folders and subfolders listed beneath your email address (shown at the top of the navigation bar on the left of the Outlook email screen).
You should regularly delete unwanted messages in your Inbox, subfolders and Sent Items folder.
For more information on propagation, see this article: Please ensure that your mail settings are correctly set.
You can have your mail and your website hosted at two different locations, but this requires correctly setting your mail server set as either local or remote.
Note: I say “currently” because it would be relatively easy for Microsoft to add support for this in later iterations of the Outlook Hotmail Connector and most IMAP implementations also have the Quota extension/capability installed which would allow Outlook to obtain the quota information for your mailbox.
Make a feature request by contacting Microsoft Support or keep your fingers crossed and it might be added in a later version or update for Outlook .
One of my previous article explains as to how to find the mailbox quota information from Outlook 2010.
Another easier way to find the information is to turn on the "Quota Information" in Outlook 2010.
The quota information shows the amount of free space that is left in your mailbox quota and the number of items you have in each folder, say Inbox, Sent Items etc as you navigate from one folder to the other. In order to switch it on, right click anywhere on the bottom Outlook 2010 bar.