The short answer is Yes, you need to upgrade.
Some people have gone to heroic lengths to get SW to run on unsupported OS releases. Here's an example for getting iBooks Author onto Snow Leopard:
http://osxdaily.com/2012/01/20/install-ibooks-author-on-mac-os-x-10-6-8-snow-leopard/
In theory, it may be possible to run Xcode 4.3.1 on Snow Leopard or even jam iOS 5.1 support into Xcode 4.2. In practice, good luck. I would never personally do it, because development is hard enough. If Xcode or your app crashes or has issues, you have to throw out all your assumptions.
You mentioned that it is your work's iDevice. Consider getting them to upgrade your Mac OS or your hardware for you.