Does Arduino sent output to TX pinout?
You haven't set pinMode for pin 1 and using pin 1 will interfere with the serial monitor (never use 0 and 1).

#define buttonUPpin = 2;
#define LedPin = 3;
int UpinStatus = LOW;


void setup()
{
    Serial.begin(9600);
    pinMode(buttonUPpin, INPUT_PULLUP);  
    pinMode(LedPin, OUTPUT);
    // digitalWrite(buttonUPpin,HIGH);  // Don't think this is right

}

void loop() 
{
    UpinStatus=digitalRead(buttonUPpin);
    if(LOW==UpinStatus)
    {
       Serial.print('A');
       digitalWrite(LedPin, LOW);  // Not sure if High or Low depends on you circuit
    }
}