Display featured images from custom post type in the homepage template

Using wp_get_archives will return a pre-defined template for the results. You need your own custom query and template to do that. Since i don’t know what exactly you are looking for, I’ll have a basic code suggestion for your situation:

// Get a list of categories
$terms = get_terms(array('taxonomy' => 'category')); 
// Run a query for each category
foreach ($terms as $term){
    $args = array(
    'post_type' => 'custom_type', 
    'posts_per_page' => '1',
    'order_by' => 'date', 
    'order' => 'ASC', 
    'cat' => $term->ID
    );
    $new_query = new WP_Query ($args);
    if ($new_query->have_posts()) {
        while($new_query->have_posts()){
            $new_query->the_post();
            // Post's title
            the_title();
            // Post's featured image
            the_post_thumbnail('thumbnail');
        }
    }
    wp_reset_postdata();
}

This will list 1 post out of each category, and output the featured image and its title. However it will make a lot of queries if you have a lot of categories. ( like 50 or so ). I suggest you use it with a cache plugin.

You can also get a list of posts and view them in a showcase, alongside their category name:

$args = array(
'post_type' => 'custom_type', // Your custom post type
'posts_per_page' => '8', // Change the number to whatever you wish
'order_by' => 'date', // Some optional sorting
'order' => 'ASC', 
);
$new_query = new WP_Query ($args);
if ($new_query->have_posts()) {
    while($new_query->have_posts()){
        $new_query->the_post();
        the_title();
        the_post_thumbnail('thumbnail');
        // Get a list of post's categories
        $categories = get_the_category($post->ID);
        foreach ($categories as $category) {
            echo $category->name;
        }
    }
}
wp_reset_postdata();