Can I personally email to a existing fiverr buyer to initiate a project with him? In this case after mailing he may start working with me by making an order.
No you can not.
On fiverr it is called spam.
And on top of that you can not message “buyers” because fiverr protects them from unsolicited messages.
This is against fiverr TOS.Dont try to do that.
If a buyer wants to work with you - they will message you.