Using Late 2013 MacBook Pro with LG Ultrafine 4K Monitor - USB-C to Thunderbolt 2?
No. You need a USB-C connection to work that display. 

You’ll spend hundreds of dollars to translate the video between formats since it basically is a small computer / device that takes power and re-encodes the signal.

Far cheaper and less hassle to resell your display (or trade / buy-sell used Macs) to get your gear on USB-C or not on USB-C.