The production is mentioned on the festival web site (click on 1967), but no detail or photo is given. Here's what I know.
Translation: Adonis (Ali Ahmed Said)
Director: Mounir Abu Dibs (the legendary Ba'albek Festival founder - more here)
Production: Ba'albek Theatre Troupe
Locations: Byblos, Deir al-Kamar, and Ba'albek
Date: 1967 -- apparently at that year's Ba'albek summer festival? Right after the 1967 war??
Cast: will post more as I find out. For now all I know is that Michel Naba'a played Hamlet. Choreographer/dancer Georgette Gebara did the choreography and played the Player Queen in the play-within-a-play.
Apparently the show enjoyed a very involved audience, as this joke repeated online testifies:
Years ago, a performance of Hamlet in Arabic took place in Byblos with Michel Naba'a in the lead role (Directed by Mounir Abou Debs in Arabic). During the scene when the Ghost appears and advises Hamlet on what to do, as he is leaving, he says to Hamlet in Arabic "LA TANSANI YA HAMLET". [Don't forget me, Hamlet.] Hamlet shrieks out "ANSAAK????". [Forget you???] Whereupon, the audience joined in : "Da KALAAM??".
(Here is "Ansaak, Da Kalaam?" [Forget you? What an idea!], the Umm Kulthum song the joke is referring to.)
But it seems not to have been new in 1967, but rather (and this would make much more sense, both war-wise and Shakespeare-quadricentennial-wise) in 1964 or earlier. A Mounir Abou Debs adaptation of Hamlet is mentioned as early as 1963 in a UNESCO report, as an example of the televised drama in Arabic that was raising the overall cultural level of Lebanese TV programming.
Another UNESCO report, this one a book-length 1981 study by Joseph Abu Rizk titled La Politique Culturelle au Liban, cites the production of Shakespeare's Hamlet (among a long list of other prestigious works) as evidence of "le niveau atteint par le theatre [libanais]." (67-68).
If you're in Lebanon at the moment (though you probably have other things on your mind), you might be able to find more info and/or some photos of the Hamlet production somewhere in here.