You can use any number of techniques for this. I would probably go with a simple expression to offset character values with a slider. Because all months do not have the same number of days this would require a few keyframes, but this would not be difficult. You would need more than one text layer. The Month names can simply be keyframed on a single layer or you could use multiple layers with a nice transition.
If you have no idea about how to write expressions or what I'm talking about please read the help files about animating text and check out how the Decoder animation preset works. You will find it in the Effects and Presets panel.
Thank you so much for this Rick. Would this be easily adjustable in duration? I mean if I do it in keyframes or multiple layers it would be painstaking if i need to change duration on it once its created?
Yes.... This is not a one click on an effect project, but it's not difficult.
You can always hire someone to write those expressions all with fancy calendar logic for the next few hundred years.... Otherwise I a gree with Rick - you are trying to see complication where probably not more than a handful of keyframes on text animators would be involved. Really not sure what would be so difficult about animating e.g. 12 month names with a per-word/per-line animator once you have typed them out...