See the documentation for ArrayList#remove(int), as in the following syntax:

list.remove(list.size() - 1)

Here is how it's implemented. elementData does a lookup on the backing array (so it can cut it loose from the array), which should be constant time (since the JVM knows the size of an object reference and the number of entries it can calculate the offset), and numMoved is 0 for this case:

public E remove(int index) {
    rangeCheck(index); // throws an exception if out of bounds

    modCount++;        // each time a structural change happens
                       // used for ConcurrentModificationExceptions

    E oldValue = elementData(index);

    int numMoved = size - index - 1;
    if (numMoved > 0)
        System.arraycopy(elementData, index+1, elementData, index,
                         numMoved);
    elementData[--size] = null; // Let gc do its work

    return oldValue;
}

You could use .remove(index) to remove the last element from a list. The index of the last element is always size() - 1.

List<String> years = new List<String>{ '2023', '2024', '2025', '2026', '2027' };
years.remove(years.size() -1 );
System.debug(years); // Print (2023, 2024, 2025, 2026)

Please note that .remove(index) returns the elment being removed, so writing years = years.remove(years.size() -1 ); would result in the following error:

Illegal assignment from String to List

Once you call .remove(index) the size of the list changes.

The for-loop idea is fine, but you should use .get(index) or the square bracket notation to get the element of index i, that's why you got

Method does not exist or incorrect signature: void str(Integer)

List<String> years = new List<String>{ '2023', '2024', '2025', '2026', '2027' };
List<String> years2 = new List<String>();
Integer lastElementIndex = years.size() - 1;
for(Integer i=0; i<lastElementIndex; i++){
   years2.add(years.get(i)); // or years2.add(years[i]);
}
System.debug(years2);

List indices go from 0 to list.size() - 1. Exceeding the upper bound results in the IndexOutOfBoundsException

list.remove(list.size() - 1);

Your test should be like below. In the test code in the original post, you are not actually invoking the method that you are trying to test.

public class UTest
{
  @Test
  public void testMultipleLast() {
     ArrayList<Integer> input = new ArrayList<Integer>(asList(1,1,3,5,7,1,5,9,1));
     ArrayList<Integer> result = new ArrayList<Integer>(asList(1,1,3,5,7,1,5,9));

     // int x = ?
     ArrayList<Integer> actual = SomeClass.removeLastOccurrence(x, input)
     assertEquals(result, actual);
  }
 }

and the removeLastOccurrence() method can do the following

if(list != null && !list.isEmpty()){
    list.remove(list.size() - 1);
}

You are getting IndexOutOfBounds because you are trying to remove an item that isn't there, so you should perform a check on your removal to stop the removal if there is nothing there:

if(array.size() > 0){
    array.remove(array.size() -1);
}

You could consider using the subList(...) method of ArrayList.