I migrated from Outlook.com for company emails to Google Apps.
How can I get the old emails on my Outlook.com to Google Apps email?
What you probably want to do is follow the same process as you would to Centralize mail from different accounts with Mail Fetcher. This will allow you to connect to your old Outlook.com account and download the emails you have there into your new Gmail account.
You'll need to use the following:
If you only want to import existing messages, you can disable Mail Fetcher after the initial import.
Your best solution is to use the Google Apps Migration Tool for Microsoft Exchange.
Despite the name, it also works with IMAP services.
It will pull the messages and folders (including the sent items). It can also do multiple accounts at one time.
If you use Mail Fetcher, you will not get the sent items.