ブロックの順番を変えるだけ。ついでに国名表示は消す。
address.php
<?php
if ( ! defined( 'ABSPATH' ) ) {
        die( '-1' );
}

$venue_id = get_the_ID();

$full_region = tribe_get_full_region( $venue_id );

?>
<span class="tribe-address">

<!-- 郵便番号 -->
<?php
// This location's postal code.
if ( tribe_get_zip( $venue_id ) ) : ?>
        <span class="tribe-postal-code"><?php echo tribe_get_zip( $venue_id ); ?></span>
<?php endif; ?>

<!-- 都道府県 -->
<?php
// This location's abbreviated region. Full region name in the element title.
if ( tribe_get_region( $venue_id ) ) : ?>
        <abbr class="tribe-region tribe-events-abbr" title="<?php esc_attr_e( $full_region ); ?>"><?php echo tribe_get_region( $venue_id ); ?></abbr>
<?php endif; ?>

<!-- 市町村区 -->
<?php
// This locations's city.
if ( tribe_get_city( $venue_id ) ) :
        if ( tribe_get_address( $venue_id ) ) : ?>
                <br>
        <?php endif; ?>
        <span class="tribe-locality"><?php echo tribe_get_city( $venue_id ); ?></span><span class="tribe-delimiter">,</span>
<?php endif; ?>

<!-- 市町村区よりあと -->
<?php
// This location's street address.
if ( tribe_get_address( $venue_id ) ) : ?>
<span class="tribe-street-address"><?php echo tribe_get_address( $venue_id ); ?></span>
        <?php if ( ! tribe_is_venue() ) : ?>
                <br>
        <?php endif; ?>
<?php endif; ?>


<!-- 国は表示しない
<?php
// This location's country.
if ( tribe_get_country( $venue_id ) ) : ?>
        <span class="tribe-country-name"><?php echo tribe_get_country( $venue_id ); ?></span>
<?php endif; ?>
 -->
</span>