I'm not surprised AE lags if you have just 4GB -- that's barely enough to keep AE and the OS running.
as Dave said you are scraping the bottom in terms of system requirements for after effects: After Effects System Requirements for Mac OS and Windows . the file size is not an indication that it should take time to process. most of the time its the opposite because it suggests that the movie is lossless i.e takes less time to render a frame. if it's compressed it will take longer to process.
but I think the big question is what exactly are you set to accomplish here? editing video scenes? this is not Ae's strong suit. you would be better off doing that in Premiere. I don't exactly understand what do you mean by "making a video like the one above" what am I seeing exactly? what exactly do you want to do?
what I am saying is that we need more information.
Try using Premiere Elements if you don't have a computer that meets the minimum specs (higher that the minimum is really necessary for Premiere Pro).
That kind of editing is not a job for After Effects.