Hi there. Well, we really only have a couple of rules here, one of which is be specific - so if you could, tell us exactly what you want us to help you with.
To get you started tho, I'll say that yes, you can use the arduino to display charge on an lcd (you will have to monitor the battery's voltage using the arduino and "graph" that). As for switching between backup and lithium, I am pretty sure you can find an ic for that task.