Is it possible to press a button from Arduino and my computer will get a key event?
The Uno is not the best board for this. You may be able to get somewhere with the Virtual USB project but there are a couple of boards that are better suited for this.

Those are the Arduino Leonardo, Micro, and Due. In those, you can use the Keyboard Library just like you are using the Serial library now. This page has an example very similar to your project:

void setup() {
  // make pin 2 an input and turn on the 
  // pullup resistor so it goes high unless
  // connected to ground:
  pinMode(2, INPUT_PULLUP);
  Keyboard.begin();
}

void loop() {
  //if the button is pressed
  if(digitalRead(2)==LOW){
    //Send the message
    Keyboard.print("Hello!");
  }
}