You can find instructions on setting up Gmail to use your Pobox address on Google's help pages. However, if you follow their default instructions, some email programs, including Outlook, will show the message as
From: firstname.lastname@example.org, on behalf of email@example.com
To avoid this, you can also tell Gmail to send mail through Pobox's SMTP servers. To make this change, click the Settings (gear icon) at the top right of the page.
1. Under Settings, click "Accounts and Import".
2. Under Send Mail As, cilck "Add another email address you own".
3. Enter your name and Pobox email address in the pop-up box. Make sure you leave "Treat as an alias." unchecked. Click "Next Step".
4. On the next step, change your settings to the following:
SMTP Server: smtp.pobox.com
Username: Your Pobox address, or Personal Domain Address
Password: Your Pobox password or App-specific password
Make sure to change the outgoing port to 587 and to select TLS.
Then click "Add Account" to finish the setup.
If these instructions do not work, please see their blog post for (hopefully) up-to-date instructions on how to set this up.