Let's answer your original question first. Compression sleeves may or may not help. I know people who swear by them, and others who say they don't make a difference. Some of those I know who use them wear them for recovery after the run, but not during the run. In my opinion, they will provide some muscle support, but minimal really. They also provide some assistance with circulation, again minimal.
Other things that may help have been alluded to in some of the other posts. Paying a little attention to your stride can make a difference, primarily planting your foot under your body rather than ahead. Don't worry too much about heel/midfoot/forefoot striking. Research has shown that injury incidence has little to do with the type of footstrike. And you probably don't know which is which or which you're doing anyway. Just work on bringing your footstrike under your body/hips. (Hard to do walking though.) Also, go slow. Just because you're "running" doesn't mean you should be sprinting. Too fast for your conditioning will contribute to the problem.
Most running injuries are "overuse" injuries, shin splints included. So you're asking more of those muscles than they are ready to give. Old, worn-out shoes can contribute. So can "overstriding" - landing with your foot well ahead of your body. Running too fast can do it too - that's when mine would show up. Muscle strengthening exercises can help (http://www.sportsinjurybulletin.com/archive/0161.htm, I use #1). Heelstrike is definitely a "straw dog", particularly considering that 70% or more of elite runners heelstrike.