Well I have a laptop from 2011. Would it work with that? And if so. Do I need to get a graphics card?
Probably.. the minimum requirements for graphics are pretty mild (though they still exist and specify an nVidia or ATI card). Your integrated graphics processor might be able to handle FFXI given the requirements are so low.
I can't remember if FFXI has a free trial available or not. I assume it does but don't hold me to that. Your best bet, should something like that exist, is to try the free trial and see what happens.