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?
Probably not. It depends on who sent you what.
I couldn't remember exact purchased date but I do remember that they opened bitcoin exchange market and I could saw bitcoin chart just after I bought it.
Forget the date. What year was that?
I remember bitcoin price jumped to nearly $1000 after they open exchange.
Who is 'they'? How do you remember that and not remember the year or what the exchange was? How did you pay them? You may have the necessary information in your bank statements.
I just want to make sure the old bitcoin has value.
There is no 'old' bitcoin or 'new' bitcoin.