I've got the same problem, Win 7 64, AMD 5200+, 3GB RAM, GTX260. I'm using software mode.
For the resolution, edit your Simcity 4 shortcut similar to this example and replace the res numbers with 1920x1080:
"G:\Games\SimCity 4 Deluxe\Apps\SimCity 4.exe" -CustomResolution:enabled -r1680x1050x32 -f -intro:off