I don't know which mail server you use. If it's gmail all you need is the user ID (normally his email address) and the password. You can access your gmail account from any computer using ID and password. You reach gmail by typing in the address bar mail.google.com
If it's yahoo, again you shouldn't have any problems. If it's another email provider I'm afraid I couldn't help you. But if you didn't sync correctly you may have had a problem with another provider that doesn't have the simple google way of entering your ID and password from wherever you might be. You may have accidentally deleted the mail account in the old phone while syncing and then you would have to reach your provider (just like mail.google.com) and then sign in.