If you want to manipulate just the array, one simple way is to copy the data in the into the array variable and then resize it.
For example
Array.Copy(arr, 1, arr, 0, arr.Length - 1);
Array.Resize(ref arr, arr.Length - 1);
ADDED
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Or as Paolo Zemek suggested
string[] arr2 = new string[listBox2.Items.Count - 1];
Array.Copy(arr, 1, arr2, 0, arr.Length - 1);