I honestly don't think so. You need Android Kitkat and your device is about 5 years old from initial release date. I am not 100% sure that CyanogenMod will correct the issue. Will it make your phone run smoother? Probably. But rooting a device is not something that should be taken lightly and if not done correctly can easily brick your device. (I knew what I was doing and once erased the entire OS from a Nexus). That being said, rooting does have its benefits but I don't think they would help in this case. Per the support page for Pokemon:
Supported devices
Android
Android 4.4 to Android 6.0.1
(Android N will not supported until the official Android release)
Preferred resolution of 720x1280 pixels (Not optimized for tablet)
Strong internet connection (Wi-Fi, 3G, or 4G)
GPS and Location Services
Intel CPUs are not supported
iOS
iPhone 5+
iOS 8+
Strong internet connection (Wi-Fi, 3G, or 4G)
GPS and Location Services
Jailbroken devices are not supported
Your device appears to not meet the display requirements either.
If you are interested in learning about rooting your device, visit the forums here and over at XDA. This will give you a better understanding of rooting. Here is a link to help:
http://forums.androidcentral.com/bionic-rooting-roms-hacks/396046-how-root-your-droid-bionic.html
However, a device upgrade would be the better route to take. Look at the marketplace here and also at sites like swappa that have great deals on devices. For instance, you may be able to get last year's model for a very good price when a new model is announced.