Two buttons, one LCD
I guess it's something with the circuit ... can you add it to your question.

Also, make sure buttonPin has the values 13 and 12 in the first and second index.

Btw, The following code is functionally identical:

void loop() 
{
    if (digitalRead(buttonPin[0]) == HIGH)
    {
       lcd.print("  Screen A ");
    }
    if (digitalRead(buttonPin[1]) == HIGH)
    {
       lcd.print("  Screen B ");
    }
}


If you want to have the loop and a flexible amount, use something like:

#define NR_OF_PINS 2

int buttonPins[NR_OF_PINS] = { 8, 9 };
String screens[NR_OF_PINS] = { "Screen A", "Screen B" };

void loop()
{
    for (int pin = 0; pin &lt; NR_OF_PINS; pin++)
    {
        if (digitalRead(buttonPins[pin]) == HIGH)
        {
            lcd.print(screens[pin]);
        }
    }
}


Update

My answer is related to the code before you edited.

Remarks:


After your edited code, you forgot to declare buttonPin:

int buttonPin[] = { buttonPin1, buttonPin2 };
You do not use long randNumber
Instead of a global variable int buttonState  you can create it locally in the loop function (although you do not need it according to my code above).

buttonState = digitalRead(buttonPin[x]);
Also, what you high likely want to add is button debouncing (check the arduino example about this), unless you add a delay after showing a screen of a considerable amount (like 100 ms or more).