You can upload to your Arduino board over Ethernet if you flash it with a special bootloader named Ariadne.

You will need an Ethernet module or shield that uses the W5100, W5200, or W5500 Ethernet controller chip. You can not use the ENC28J60 Ethernet controller with Ariadne.

You will need an ISP programmer to burn the Ariadne bootloader to your Arduino board. If you don't own an ISP programmer, you can use a spare Arduino board as an "Arduino as ISP" programmer. If you're going to purchase an ISP programmer, I recommend the 6 pin USBasp. The USBasp works well with the Arduino IDE and is very reasonably priced. You can buy them on eBay.