Apple 30" Cinema Display only supports two modes: 1280x800 60Hz and 2560x1600 60Hz.
The latter requires a dual link DVI connection. You'll need a HDMI to DisplayPort adapter with a DisplayPort to dual link DVI adapter if you want 2560x1600 from HDMI.
Modern game consoles usually support HDMI which should be compatible with single link DVI which can get you 1280x800. If your game console supports arbitrary modes, then 1280x800 should be possible.
If your game console doesn't support the modes that the display supports (I don't know if any game consoles do), then you'll need a scaler that can scale the output from the game console to the mode supported by the display. I don't know if a scaler exists for this use case. The scaler may also need to be able to change the timing since the 60Hz used by the display might not be the same as the 60Hz of the HDMI output of the game console. Some scalers only support specific output modes. Also, the Cinema Display only supports RGB so you can't use YCbCr 4:4:4, 4:2:2, or 4:2:0. The Cinema Display is 16:10 instead of the usual 16:9 so the scaler may need to add black bars on the top and bottom if it cannot get a 16:10 mode from the game console.
Do consoles require HDCP? The newer Apple 30" Cinema Displays support HDCP but there are older ones that do not.
