Can the Arduino use this i2c LCD Panel
Unfortunately you cannot control that display using that library; the library expects to be connected to a HD44780-compatible driver but the display uses a fully custom control scheme. You may be able to adapt the library to the scheme, but I have not looked at that specifically.

I2C itself only requires 3 connections: SDA, serial data; SCL serial clock; GND, ground reference. Since the display is not powered separately it also requires power, in this case 5V. Since the display can only communicate at 100kHz you don't need to be too terribly concerned about cable quality; Cat3 or even telephone cable is acceptable as long as you don't exceed a few feet in length. Don't forget the external pullup resistors though, on SDA and SCL.