How to add css class attributes to the anchor element inside the links of a given menu by theme function
I am probably looking at this upside down or backwards but, are you rendering this menu yourself, eg, you are calling theme('links', $variables) somewhere and can easily modify $variables?

If so, a quick look at the theme_links() function itself reveals the following comment...

/**
 * Returns HTML for a set of links.
 *
 * @param $variables
 *   An associative array containing:
 *   - links: An associative array of links to be themed. The key for each link
 *     is used as its CSS class. Each link should be itself an array, with the
 *     following elements:
 *     - title: The link text.
 *     - href: The link URL. If omitted, the 'title' is shown as a plain text
 *       item in the links list.
 *     - html: (optional) Whether or not 'title' is HTML. If set, the title
 *       will not be passed through check_plain().
 *     - attributes: (optional) Attributes for the anchor, or for the &lt;span&gt;
 *       tag used in its place if no 'href' is supplied. If element 'class' is
 *       included, it must be an array of one or more class names.
 *     If the 'href' element is supplied, the entire link array is passed to
 *     l() as its $options parameter.
 *   - attributes: A keyed array of attributes for the UL containing the
 *     list of links.
 *
 *  (rest of comment deleted)
 */


showing that what you need to do is put in an attributes key with each link, as, further on down in the code, when it is creating the link itself, it does, as the comment above also mentions, adds in the full $link as the $options parameter, eg...

$output .= l($link['title'], $link['href'], $link);


So, if you create an attributes key that points to an array with a key of class for whatever link needs a special class, and that key itself points to an array containing that name, this class will be added to the anchor.