Add styling to an element on hover to the parent element

If what you want is that when passing the mousethrough the divapply the style text-decoration: underline;, try the following codigo.

#addNewColumn {
    width: 200px;
    height: 300px;
    border: 1px solid green;
    border-radius: 5px;
    border-width: 2px;
    margin: 20px;
    cursor: pointer;
}
#addNewColumn p {
    text-align: center;
}
#addNewColumn:hover {
    text-decoration: underline;
}

<div id="addNewColumn">    
        <p id="newColLink">   Underline on hover </p>
    </div>

Applies from CSSthe hoverto the parent #addNewColumn:hoverbut the rule CSSwill apply to the child (p)

#addNewColumn{
width: 200px; 
height: 300px; 
border: 1px dashed green;
}
#newColLink{
text-align: center;
}
#addNewColumn:hover p{
	text-decoration: underline;
}

<div id="addNewColumn" >    
    <p id="newColLink"> Underline on hover </p>
</div>

To apply what you want by hovering on an element BUT assigning the styles to another element you could do the following:

#addNewColumn:hover p{
  text-decoration: underline
}

<div id="addNewColumn"  style="width: 200px; height: 300px; border: 1px dashed green; border-radius: 5px; border-width: 2px; margin: 20px; cursor: pointer">    
    <p id="newColLink"   style="text-align: center;">   Underline on hover </p>
</div>

With the selector #addNewColumn:hover pyou are saying that all the elements <p>that are inside the selector are div#addNewColumnapplied a certain style when hovering the mouse over it.

NOTE: inline styles (as you are using them) are bad code practice, you should have an external stylesheet and link it to the HTML via the tag<link>

You leave your body as you have it:

<div id="addNewColumn"  style="width: 200px; height: 300px; border: 1px dashed green; border-radius: 5px; border-width: 2px; margin: 20px; cursor: pointer">    
    <p id="newColLink"   style="text-align: center;">  
        Underline on hover
    </p>
</div>

In your css file that you will have linked to your html, add the following:

    #addNewColumn:hover p {
        text-decoration: underline;
    }