I've the next CSS in attempt to produce a rollover effect:

#arrow {
    position:absolute;
    left:540px;
    top:150px;
    width:220px;
    height:120px;
}
#arrow a{
    display: block;
    width:220px;
    height:120px;
    text-decoration: none;
    background: url("images/arrow.jpg");
    background-position: 0 -120px;
    }
#arrow a:hover {
    background-position: 0 0;
    }

And then, the div tag:

<div id="arrow">
    <a href="/arrow/arrow.htm"></a>
</div>

However, when my mouse will roll over the look, it does not alter the background position from the link. Ideas?

Add background-repeat: no-repeat to #arrow a.

It's working, however your background is repeating therefore it may seem like it isn't working whatsoever. So either set no-repeat or the backdrop-repeat or simply remove the url setting after which add it on hove only