How can I contact imac 2013 late to MacBook Air 2013 with mini display cabel??

That setup would would not work at all. Rudegar missed a couple of points:

1. You're trying to use the MacBook Air as a monitor. The 2013 iMac can act as a Target Display Mode display – in some very restrictive cases that are even more restrictive now, than when it came out. The MacBook Air never offered the ability to act as a Target Display Mode display of any kind.
2. Even if you were using the Late 2013 iMac as a second monitor for the Air, and you met the OS restrictions, the Late 2013 iMac cannot accept Mini DisplayPort input. It requires Thunderbolt input. While the connectors on a Mini DisplayPort cable are the same as on a Thunderbolt one, you need a Thunderbolt cable, to assure that the cable can reliably carry data at full Thunderbolt speed.

it would not work at all unless both macs is running pretty old macOS versions

Use your iMac as a display with target display mode - Apple Support