There are no good cards for 1080p gaming for that price unless you go USED.
If possible:
1) GTX750Ti performance or better
https://www.techpowerup.com/reviews/NVIDIA/GeForce_GTX_750_Ti/25.html
For example, an HD7850 is an older card but performance is similar to a newer GTX750Ti.
2) 2GB video memory or greater
So you could look for some of those cards used around the level of the GTX750Ti. The GTX560Ti is probably similar, though may only have 1GB of VRAM.
Other:
GTA 4 runs like crap on my current computer (i7-3770K + GTX680) and ran crap on the other computers I've tried. I've spent countless hours trying to fix the issues and finally just gave up. It often works fine for a time then just runs like crap at others.
Saints Row 3/4 run much better if that's something you'd like.