If you have shared mailboxes added to your Outlook and they are all showing New Mail Desktop Alerts when new email comes in you can set a rule to just show alerts for a particular mailbox.

There is no direct setting for this in Outlook, but you can achieve this by disabling the generic option to display the New Mail Desktop Alert and then create a rule to display it again when mail arrives via your specified account(s).

Disabling New Mail Desktop Alert

The first step of the process is to disable the generic option to always display the New Mail Desktop Alert:

• Outlook 2010 and Outlook 2013 File-> Options-> Mail-> section: Message arrival-> option: Display a Desktop Alert

Creating a New Mail Desktop Alert rule for specific accounts

The next step is to create a rule to display the New Mail Desktop Alert again but only for the accounts that you select.

You need to create the following rule:

1. Open the Rules and Alerts dialogue by going to  File > Manage Rules & Alerts;                                                       
2. When you see the "Apply changes to this folder" drop down list at the top, make sure that the account for which you want to see alerts is selected here;
    
3. Click New Rule…;
4. In the Start from a blank rule section, verify that Apply rule on message I receive is selected;
5. Press Next to go to the Conditions screen;
6. Select through the specified account;
7. At the bottom, click on specified and then select the account for which you want to see the alert;
8. Press Next;
9. Select the action Display a Desktop Alert;
10. Press Finish to complete the rule.

The entire rule will now read as follows: