Try on of these and see if these help.
Removing and re-adding your account from the phone. Check out this link for the steps: Mobile Outlook.com
(Phone may have to restart to work)
Ensure that the mail server settings are properly configured. Refer to this link for the values: Send and receive
Hotmail from an email app.
Ensure you phone has the most recent firmware update. You can do this but check with the phone maker. ( They usually have a downloader or something you can you to check from a PC)
If your android firmware is lower than 2.3 the outlook app will not work. You can check your firmware version by going to "Settings" and "About Phone" and " Firmware Version".
If it has 2.2.(whatever) the app won't work. Been looking around and it seems like this is what the most common presences is.
Hopefully Microsoft is working on this and it will be fixed on the next update. People with the older versions of Android are pretty much SOL'd and will have to reply on 3rd party email apps or linking the email account
to another providers email account (i.e. Hotmail link to Gmail so your emails go to the Gmail account