That MacBook is a Late 2006 model. Apple's specs say that it can only use 2GB of RAM but other users have reported being able to use up to 3GB.
You can bring the operating system up to version 10.7.5 (Lion.) Mountain Lion (10.8) will not run on it.
If you have Mac OS X 10.6.8 as your stats say you do, your MacBook will work just fine with your iPhone 5. You will want to download the latest version of iTunes by going here or using Software Update (under your Apple menu.)
If you decide you want Lion anyway, you can get it by contacting Apple sales by phone and purchasing a download code which you then use in the Mac app store. In the USA the phone number is (800) MY-APPLE. If you are elsewhere, go to store.apple.com and look in the upper right corner for your country's sales phone number.
Best of luck.