So looking online at system spec requirements it is a pretty simple game. 2.3GHz processor (assuming single core), 1GB of RAM, 100MB drive space.
That said, from what I can tell it is largely flash based, and even the log-in page crashed both IE and FireFox... so we are not exactly talking about a game that is optimized or well written. My guess is that your old Core2Duo is the cause of any lag as it just barely eeks over the 2.3GHz requirement. Also, systems that old tend to only have 1-2GB of RAM by default, which means that you probably don't have a full 1GB of ram to dedicate to this game (windows and a web browser alone will eat 4GB).
So here is the rub:
1) adding a GPU will not help with this game, plus being a mini-tower you would need a SFF GPU, and likely a small (better than stock) PSU to power it.
2) getting your system up to 4-8GB of ram would help... but DDR2 is somewhat expensive, and your system is not really worth sinking that much money into it. If you can find working used RAM for cheap or free (check local listings on craig's list, or see if there is a computer co-op in your area that will sell it to you for cheap) then great... otherwise don't waste your money.
3) what you really need is a new CPU... and while you can find faster Core2 chips for cheap, it is a lot of effort to swap out the CPU in a pre-built system, and you have to be sure to get one that is specifically compatible with your motherboard (which can be difficult).
4) keep in mind that this is not a high-end performance oriented game... the lag issues could very well be on their end rather than yours. Chat with other players in your physical area and see if they experience similar issues. If the problem is server-side then no amount of upgrades will help you.
I guess what I am saying is that this is a very simple game, and if your computer cannot play it then it is time for a newer box rather than throwing money at your current rig. Thankfully $3-500 for a new PC can get you quite a lot of computer these days. For this game in specific I would just look for a PC with a dual core processor that has the highest clock rate possible (at least 3GHz) and at least 4GB of RAM. Should not be hard to find, and it should not be expensive, and it will be much faster than your current 7 year old computer.