What's the max resolution for mini DisplayPort connected to a 2013 MacBook Air?
You can get full resolution with mini DisplayPort to DisplayPort or mini DisplayPort to Dual-link DVI.  Dual-link DVI supports up to 2560x1600 pixels resolution.  DisplayPort supports that resolution or even higher.

You will not get full resolution with mini DisplayPort to DVI.  DVI is limited to 1920x1200 or something similar.  You usually won't get full resolution with mini DisplayPort to VGA.

I do not know whether you get full resolution with mini DisplayPort to HDMI.  I think it depends upon your graphics card.

Bottom line. I would recommend using mini DisplayPort to DisplayPort.