In the searchTextbox subscribe to the TextChanged event. Within that listener use
int searchLength = searchTextbox.Text.Length;
listbox2.Items.Clear();
for (int i=0; i<listbox1.items.count;i++)
if( listbox1.Items[i].ToString().substring(0,searchLength) == searchTextbox.Text)
listbox2.Items.Add(listbox1.Items[i].ToString());
Please note my first submission I accidentally left out the ToString() within the if-statement. The above code is now correct, It had previously read
if( listbox1.Items[i].substring(0,searchLength) == searchTextbox.Text)