I lived in Kuwait for 2 years and loved it. Joined the Radisson for the gym/pool, went to Zumba at a place in Salmiya. Can't remember the name of the place but they did loads of other courses as well - languages, IT etc so studying could be an option for you. There are a number of charities you could get involved with - not that I ever did. Then the usual M.E. things like shopping, coffee........., walks along the corniche in the cooler weather (before/after a trip to Starbucks). There are quite a few parties/events so it isn't too boring.
I lived there for 2 years before moving to Dubai. I lived in Salmiya too, just across from Marina Mall. There isn't a lot to do in your free time to be honest....I joined a aqua-aerobics class and a swim training class and by the time I left there I was so fit it was excellent !!!! What nationality are you as there are coffee mornings you can go to. There is also lots of music and art sessions that you can visit in the evening.