I don't want to put Alice off (I work at QU and so I should be loyal) but I've checked further and the Arabic taught is definitely Classical which, as a Qatari friend remarked, is difficult for many Arabs: that is, they understand it but can't speak it well. Nevertheless, if you take a positive approach to language learning, you should be able to look at the classical and modern Gulf Arabics side by side and enjoy noticing the differences. Although you won't be immersed in Arabic, you'll certainly be surrounded by it. With due respect to Aisha, my students tell me that chatting with the Arabic learners in Arabic isn't easy, but yes, it does happen.
Go for it! Just don't expect to be drawn into a millieu in which you can 'acquire' Arabic 'naturally', or hang out with female Arab friends in the evenings. You'll maybe get invited to weddings, but homes are much less likely. For that reason, don't go thinking that spending time with other Westerners will be some sort of cop-out. Do check out Western social outlets (see other threads) for the sake of sanity. Besides, you'll find some of them (though not me) speak a bit of Arabic and will be helpful.
I've been here five years working for and with Qataris and I like it, but it is a different culture. Enjoy it and explore it, but don't expect to find a home from home, any more than a good Qatari girl could expect to fully fit in with her party-going or modern-intellectually minded British peers. Religious and family values are over-riding here in a way that non-Moslems (and even some born Moslems) cannot grasp.
Good Luck!