UPDATE
There are some mistakes:
The value of color is a name, 6 or 3 hexadecimal number, or RGB/RGBa, or HSL/HSLa 
I've never seen {color:links nav} as a value before... is this actually working?
As @torazaburo and @Ralph.M pointed out, your divs qualify as first-childas well as first-of-type of a only, so your .update is not the direct ancestor (i.e. parent) of any div. Therefore you need more specificity by going through each level of the hierarchy.
div.update.nav  
a 
div 
Try this:
  .update a:first-child div:first-of-type
Note: In your circumstance first-child and first-of-type works the same, the difference I can think that might be of concern is if you use first-child and nth-child the n-th variable counts differently than nth-type-of.
The :first-of-type selector in CSS allows you to target the first occurrence of an element within its container. 
.update div {
  width: 100%;
  height: 25%;
  border-top: 2px dashed red;  /* this is a color name */
  padding: 5px;
  background-color: rgb(0,0,0);  /* this is RGB */
  cursor: pointer;
  color: rgba(250,250,250,.7);  /* this is RGBa */
}
.update a:first-of-type div:first-of-type {
  border: none;
}
.update div:hover {
  color: hsl(240, 60%, 50%);  /* this is HSL */
}
.update div:hover > .symbol {
  color: hsla(324, 100%, 50%, .8);  /* this is HSLa */
}
<div class="nav update">
  <a>
    <div>
      <div class="symbol">×</div>update 1
    </div>
  </a>
  <a>
    <div>
      <div class="symbol">×</div>update 2
    </div>
  </a>
  <a>
    <div>
      <div class="symbol">×</div>update 3
    </div>
  </a>
  <a>
    <div>
      <div class="symbol">×</div>update 4
    </div>
  </a>
</div>
 
 
See ARTICLE