New iPhone getting old emails

If your email account was set up as IMAP as soon as you log into the account your messages should download. If it was set up as POP you need to restore a backup of the old phone. See-->Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

Add the email account that you had. That's it.

Add an email account to your iPhone, iPad, or iPod touch

1. Go to Settings > Passwords & Accounts, then tap Add Account.
2. Tap your email provider.
3. Enter your email address and password.
4. Tap Next and wait for Mail to verify your account.
5. Choose information from your email account, like Contacts or Calendars, that you want to see on your device.
6. Tap Save.

Add an email account to your iPhone, iPad, or iPod touch - Apple ...

https://support.apple.com/en-us/HT201320