It's not about the RAM. It's about the GPU chip itself. I'm not sure how the 945 and it's revisions are exactly performance-wise, but the integrated GPUs have really never been able to run the OpenGL 2 plug-in with high settings, and it seems getting the OpenGL(1) plug-in running with decent settings is hard for some users too.
Try the Direct3D one next, but it's not quite as accurate. Final Fantasy games seem to have some visual incorrectness in it, for example.
If all else fails, use P.E.Op.S. software plug-in. The visuals may not look as great, but it will be plenty fast and more accurate, and with a CPU like that, you should be able to play with the higher stretching filters (maybe even the HQX2 or HQX3 filters).
As for PCSX2, I can comfortably say forget about it. The CPU will bottleneck it too much. I had a decent GPU (at the time, quite a powerful GPU, a 6800GS AGP running as a 6800 Ultra), and even that paired with a Pentium 4 2.8GHz Hyper-threading struggled to get more than ~15FPS average (between ~10FPS-25FPS) in Final Fantasy X, which is one of the easier 3D games to emulate.
If you're wanting to buy a GPU, find out what expansion slots you PC has. Given the chipset, I doubt it's AGP, so it's either PCI, PCI Express x1, or PCI Express x16. If it only has just standard PCI slots, or PCI Express x1 slots (the really short ones), in other words, if it lacks a PCI Express x16 or AGP slot, forget about it. Otherwise, the cards will be too costly for their performance, and the higher performing cards aren't available in anything but PCI Express x16.
Edit: Adding any decent GPU will really let you use the OpenGL plug-in and pretty high settings for anything. It'll also balance your PC and give it some gaming capabilities, even if it is older and a bit long in the tooth for the latest stuff. If you have some money you don't mind dropping, and it has a slot that's worthwhile, Id say go for it. Just keep it to a mid-range card for sake of the likely basic PSU in the system.
Also note that the 945 chipset did support some Core 2 CPUs (not the Penryns, though, and I'm not sure of any quad core CPUs). That chipset was officially made for the Pentium D, but the chipset did run the first generation Core 2s after a BIOS update was made. They just didn't overclock, well, much at all on that chipset (since, as I said, the Core 2 came after the 945 chipset, and support was later added), but they still ran them stock, which is better than any Pentium 4/D. If you can make sure the BIOS of that board has support for them, that might be a good upgrade alongside a GPU upgrade.