I've been residents of Istanbul and Beirut and I would say that it is beautiful time of the year to go to Lebanon.
What would you do for one week? Well you would first of all arrive, probably relax the first day, enjoy the beach, go to a beach club, get stuffed to the gills on Lebanese food, enjoy evenings until 6am at the clubs on Rue Monot in Achrafieh, stroll amongst the streets of downtown Beirut (link provided in one fo the earlier replies) have a coffee or ice cream during the day and a nice meal at night. The atmosphere in the downtown district is something special!
You could do a day trip to Baalbeck - one of the best preserved set of Roman ruins in the world - and see what the Bekaa valley is like.
On the way to Baalback you could stop for lunch in Zahleh, a river valley town where you can have a superb meal on a long table with white tablecloths in an outdoor restaurant. Man I wish I could be there right now...
You could also go to the Cedars region and perhaps up north to the town of Tripoli but I don't recommend that you go to Tripoli.
I would think that you'd have an excellent time in Lebanon and if I can provide any further information I would be only too happy to do so, as would the other Lebanese members on this forum I'm sure.
Now as for Istanbul, I can't play that off either because Istanbul is an absolutely beautiful city, second in my mind only to Beirut but that is because I am obviously biased. I suspect however that your stay in Istanbul would most be consumed with tourist attractions such as the Blue Mosque, Hagia Sophia, Topkapi Palace and Museum as well as a couple of other things. It is also a very fun city...I spent two fo the best years of my life there...so I think it depends on what kind of a vacation you want to have. I suspect that you would have a slightly faster paced vacation in Istanbul than you would in Beirut.
PARIS, FRANCE...THE BEIRUT OF EUROPE.