Just login to your new Google account and your old email account in a desktop email app like Outlook on a PC or Mail on Mac. But, there's a workaround-the same workaround to export your emails from most email accounts.
There's actually no built-in way to import emails on your own to Gmail.
The best way to import emails to Gmail is to drag-and-drop the messages in
Just sync the emails to your mail app, then jump to the next step. You'll find similar options in the Calendar and Contacts apps as well. Open Mail, select Mailbox, then select Export Mailbox and follow the directions. On a Mac or PC, open the File menu, select Open & Export, then choose Import/Export and select the correct options for your files. That's the easiest way to export iCloud,, Yahoo!, and other standard email, contacts, and calendar accounts. Have data from another email account that you want to import into Google? Often, the best way to do that is to sync your emails to a desktop email app, and then export them. Now you're ready to import everything to your new Google account. Click the link, login again, and you can download a zip file of all your exported data. Exporting all your data can take a while, so Google will email you when everything's done.