How to connect a MacBook Air to a monitor

You can use a display up to 3840x2160 (4K), and connect it with a mini-DisplayPort (MAC end) to DisplayPort, HDMI or VGA cable or adapter (in that order of preference) depending on the input port(s) of the display.

Use an external display with your MacBook Air - Apple Support describes your options.

https://www.amazon.com/AmazonBasics-Mini-DisplayPort-Display-Cable/dp/B013PWQPFS/

TB2 and mini display port are the same physical plug. If you need to carry more than just AV (for example, the monitor has a built-in WebCam or you also need to connect peripherals to other ports on the monitor) you would need to have a TB2 display, and since that has been replaced by TB3 (which has a plug identical to and compatible with USB-C) I’m not sure you’ll find one these days.