If I could retrieve this so called "private key" or whatever they emailed me, I probably be able to restore old bitcoin and could cashing it out. Am I correct?
To make sure that you are not bluffing. It would be better to describe the visuals you have received in your email. Private keys is a 256 bit number represented using hexadecimal format which composed of characters 0-9 and A-F, and it is not usually seen by users to avoid getting compromised. Receiving priv key by email would is indeed a rare case LOL.
Also, retrieving a private key is next to impossible. The vast keyspace nature of private keys is designed to make bruteforcing infeasible, if ever you attempt such kind of activity, it will take an enormous computing power before finding that particular private key.