This steps are not authorized by Microsoft and probably even not allowed, but still this is only option to gain access to Exchange Online Shared mailbox from Windows Phone device with push delivery.

1. Install Microsoft Online Services Sign-In Assistant http://www.microsoft.com/en-us/download/details.aspx?id=41950
2. Install Azure AD PowerShell module from http://go.microsoft.com/fwlink/p/?linkid=236297
3. Start PowerShell and run these commands:
   1. Load Azure AD Modules:

      (get-item C:\Windows\System32\WindowsPowerShell\v1.0\Modules\MSOnline\Microsoft.Online.Administration.Automation.PSModule.dll).VersionInfo.FileVersion

   2. Log in as administrator:

      $msolcred = get-credential
      connect-msolservice -credential $msolcred

   3. Reset password on shared mailbox:

      Get-MsolUser -UserPrincipalName "sharedmailbox@account.onmicrosoft.com" | Set-MsolUserPassword

      Running this command will reset password to this inbox to temporary password that will be printed to stdout.

4. Use this temporary password to log into the shared mailbox using OWA on http://mail.office365.com
5. When you log in, you have to change password to new one, set any password you want.
   1. After you sucessfully change to password you will be redirected to page that states that this account does not have access to that inbox, this is correct as this is shared mailbox that is not allowed to be used directly to access inbox, but Outlook Anywhere from Windows Phone device is allowed.
6. Create new account on your Windows Phone device using shared mailbox address with this password and you have full access to this inbox even with push notifications.