JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="//netdna.bootstrapcdn.com/twitter-bootstrap/2.3.1/css/bootstrap-combined.min.css">
<script src="//netdna.bootstrapcdn.com/twitter-bootstrap/2.3.1/js/bootstrap.min.js"></script>
<div class="btn-group">
    <a class="btn dropdown-toggle" data-toggle="dropdown" href="#">
	    Inch
	</a>
    <ul class="dropdown-menu">
        <li>1</li>
        <li>2</li>
        <li>3</li>
        <li>4</li>
        <li>5</li>
        <li>6</li>
        <li>7</li>
        <li>8</li>
        <li>9</li>
        <li>10</li>
        <li>11</li>
        <li>12</li>
        <li>13</li>
        <li>14</li>
        <li>15</li>
        <li>16</li>
        <li>17</li>
        <li>18</li>
        <li>19</li>
    </ul>
</div>

CSS

ul{
    list-style-type:none;
    max-height:160px;
    padding-top:10px;
    overflow:auto;
    text-align:center;
}

ul li{
    height:24px;
    line-height:24px;
}

.selected{
    background:#ff0000;
    color:#fff;
}

JavaScript

// scrollTo plugin 
$.fn.scrollTo = function( target, options, callback ){
  if(typeof options == 'function' && arguments.length == 2){ callback = options; options = target; }
  var settings = $.extend({
    scrollTarget  : target,
    offsetTop     : 100,
    duration      : 0,
    easing        : 'linear'
  }, options);
  return this.each(function(){
    var scrollPane = $(this);
    var scrollTarget = (typeof settings.scrollTarget == "number") ? settings.scrollTarget : $(settings.scrollTarget);
    var scrollY = (typeof scrollTarget == "number") ? scrollTarget : scrollTarget.offset().top + scrollPane.scrollTop() - parseInt(settings.offsetTop);
    scrollPane.animate({scrollTop : scrollY }, parseInt(settings.duration), settings.easing, function(){
      if (typeof callback == 'function') { callback.call(this); }
    });
  });
}


//My code
//The function that is listing the the mouse
jQuery(".btn-group .dropdown-menu li").mouseover(function() {
		console.log('mousie')
	    jQuery(".btn-group .dropdown-menu li").removeClass('selected');
	    jQuery(this).addClass('selected');
})	

//What to do when the keyboard is pressed
jQuery(".btn-group").keydown(function(e) {
    if (e.keyCode == 38) { // up
        console.log('keyup pressed');
        var selected = jQuery('.selected');
        jQuery(".btn-group .dropdown-menu li").removeClass('selected');
        if (selected.prev().length == 0) {
            selected.siblings().last().addClass('selected');
        } else {
            selected.prev().addClass('selected');
            jQuery('.btn-group .dropdown-menu').scrollTo('.selected');
        }
    }
    if (e.keyCode == 40) { // down
        console.log('keydown');
        var selected = jQuery('.selected');
        jQuery(".btn-group .dropdown-menu li").removeClass('selected');
        if (selected.next().length == 0) {
            selected.siblings().first().addClass('selected');
        } else {
            selected.next().addClass('selected');
           ...