With most 5770s it wouldn't reach the RAM slots, and even if it does, you'll just have to lift up the GPU if you want to upgrade the RAM. Unless it has a thick backplate (The only one I could think of the the Asus CuCore IIRC), you'll be fine with it physically.
what specs does your 580 have though?
Core i3 550, 4GB RAM, 1TB HDD, and I just upgraded the stock 300W PSU to a 550. Now since you answered that for me, maybe you can help me with this. I know that I have 1 PCIe x16 slot but is it a 2.0 or a 2.1? I would assume that since I'm using an i3 that the slot is a 2.1. Is that correct?