Vintage Omega watch / Could you tell me more about it?

Looks like a great heirloom piece and it looks entire legit to me. FYI, it was already an old watch when your grandfather received it in the 1970s. The plain case-back is correct, but there will be more information inside. The back will pop off if you insert something sharp between the case-back and mid-case. If you look closely you will probably see a small notch somewhere around the edge. If you want to wear it, the watch should receive a service from a qualified watchmaker, since the oils will have dried up over time.