If you get out of the tourist areas, you'll find a lot less English speakers. In the Christian areas of Damascus and Aleppo, a lot of people can speak English (and some speak other European languages as well), so avoid these areas if you want to immerse yourself in Arabic.
I can speak and write some Arabic and I never had any problems using it in Syria. People always responded in Arabic. If you befriend some locals, you'll probably be invited to visit their family (you are a man, so you'll mostly meet the male relatives) and you'll get a good opportunity to practise. Don't refuse an invitation for a coffee or a tea. Syrians are really warm-hearted and will want to know all about you.
The first time I went to Syria, I could barely speak Arabic (only the numbers and basic greetings). I was invited by the Syrian family of my best friend and I spent a whole month in the country. After three weeks, I could understand quite a lot of things because EVERYBODY kept on talking to me even though I could not understand anything at first. It was Arabic 24/24, 7/7, and I had no opportunity to use my mother tongue or English because no-one knew another language than Arabic (my best friend was in another village with other family). It was quite frustrating the first few days but it gradually became fun as I became the one whom everyone could rely on to keep their secrets :-)))
I'm still pretty crap when it comes to using the language myself but I can now get on in any Arabic-speaking country... and like #2 said : knowing Arabic REALLY opens doors!!!
So GET OUT of the tourist areas and go to the villages, and meet locals... It's a really rewarding experience!