前の回答をちょっと手直し｡

<select name="begin" size="1">
<%
 for(int j=1;j<nens.length;j++){
%>
 <option value="<%="nen"+j %>"><%=nens[j] %></option>
<%
 }
%>
</select><br>
<!-- ここまで､質問文 -->

<!-- ここから､回答 -->
<select name="finish" size="1"></select> 

<script>
window.addEventListener("DOMContentLoaded", function() {
 var begin = document.querySelector('select[name="begin"]');
 var finish = document.querySelector('select[name="finish"]');
 begin.addEventListener("change", function() {
 var s = "";
 var selected = false;
 Array.prototype.forEach.call(begin.options, function(opt) {
 if (opt.selected) {
 selected = true;
 }
 if (selected) {
 s += '<option value="' + opt.value + '">' + opt.innerHTML + '</option>';
 }
 })
 finish.innerHTML = s;
 });
});
</script>