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>

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