Offset WP_Query by negative 1

I couldn’t understand your question. exactly

• Do you want next, prev post pagination?

  or

• You want to see next and prev posts titles or other content?

Actually you placed an incorrect value to offset. You have to do something like this.

// Initialize where to start the post from, 0 is most recent post
$init_count = 0;

// Get the current page integer
$page = get_query_var( 'paged' ) ? get_query_var( 'paged' ) : 1;

// And the simple formula for offset is this
$offset = ( $page - 1 ) * $init_count;

//Now you should use your query like this:

<?php $upcoming_events_args = array(
    'post_type' => 'event',
    'meta_key' => 'start_date', // name of custom field
    'orderby' => 'meta_value_num',
    'order' => 'ASC',
    'posts_per_page' => '3',
    'number'     =>  $init_count,
    'page'       =>  $page,
    'offset'     =>  $offset,
    'meta_query' => array(
    array(
        'key' => 'start_date',
        'value' => $date_of_page,
        'compare' => '>=',
        'type' => 'DATE'
      ))
    );

And by placing this code after end of the loop, will show your next and previous post.

<ul class="pagination">
    <li id="previous-posts">
        <?php previous_posts_link( '<< Previous Posts', $custom_query->max_num_pages ); ?>
    </li>
    <li id="next-posts">
        <?php next_posts_link( 'Next Posts >>', $custom_query->max_num_pages ); ?>
    </li>
</ul>