If you have permissions to a different mailbox, you can switch to that mailbox view in Office 365 Outlook.  Here are the steps:

1.  Log into your email box at outlook.office.com

2.  Click on your initials in the upper right hand corner.  Hint:  they might be backwards.

3.  Click Open Another Mailbox.

4.  Type in the email address for the account you want to switch to and click Open.

That's it!  You should now be attached to your secondary mailbox.

Sending an email from the other Inbox you can change the FROM to reflect the Inbox your in.

1. Click on New mail and then click on Options at the top.

2.  Check the box for Show From, then when you click From, you will see a list of Mailboxes you have permission to send mail from it.

That is it, now the email will show that it is from that other Mailbox and not from you.