How do I sense whether the input signal is GND or OPEN?

I'm working on a project that would require my board to read whether an input has continuity with ground, or is open.

How do I use Atmega2560-16AU's I/O Pin as an input so that I can tell whether the input signal is GND or OPEN?

Pull-up resistor ~10kOhm, that is it. You can use internal built in resistor but its value is bit higher so more sensitive for a noise.

Is it solved by attaching a pull-up resistor to VCC outside the sensing pin?

It's solved by connecting pull-up resister between sensing pin and Vcc or by enabling internal pull-up for sensing pin:

pinMode(Pin, INPUT_PULLUP);

Exactly. Same circuit as you have button from MCU's pin to the ground.

  1. Input is grounded - small current throughout pull-up resistor as it is VCC - GND connected, reading provides 0.
  2. Input is opened - very small current throughout pull-up resistor from VCC to internal circuit of MCU's input, on input is VCC, reading provides 1.
    Pull-up ensures stable state if it's open. Resistor must be small enough to provide VCC on MCU's input reliable without a noise and big enough that the current cross resistor will be so small to not represent a significant load for the circuit. Let say anything between 1k and 50k will work. 10k is good choice for ATmega.
    There are tons of schematics and pictures on the internet:
    https://www.google.com/search?q=pull+up+resistor&hl=en&biw=1366&bih=677&tbm=isch&source=iu&ictx=1&fir=Ow8divbNdRykVM%2CtJi_0j9QlH6hZM%2C_&vet=1&usg=AI4_-kScwgSLsUX3xdT5RfQXQVZ-resFZA&sa=X&ved=2ahUKEwiGx_Slg6ztAhVBzaQKHSGiDNQQ_h16BAgKEAk#imgrc=Ow8divbNdRykVM