Hello JenBryer,
Thank you for using Apple Support Communities. That's one excellent question to ask. Apple does have a public resource that explains how to import your existing email messages to your iCloud account. Click here for details: Import existing email messages to your custom iCloud Mail address on iCloud.com - Apple Support
It also states the following:
If you add a custom email domain you already own to iCloud Mail, you can use iCloud.com to import existing email messages from your previous email provider (not available for all email providers).
Although this resource does show you how to import your emails, the best approach to take is to confirm with Network Solutions if they allow the import of their email messages to occur. That way you avoid any possible data loss during the import. Just in case, you can reach them here: Contact Us | Networksolutions.com
Thank you for using Apple Support Communities. Take care.