This covers the steps to send an email from a different mailbox. You will need permissions to that mailbox to achieve this.
To send as or on behalf of emails:
1.On the Home tab, select New Email.
2. Click the 
box and select Other E-mail Address... from the drop down.
Note: The From field is hidden by default. If the From field is hidden, select Options from within the message you are composing, then select From.
3. Within the Send From Other E-Mail Addresses window, click From to select the email address of the account you want to send as or send on behalf of.
Note: When you send an email with more than one active email in your Outlook, ALWAYS check the From area as you might send the message from the wrong account!