How can I cycle posts with a #next and #previous script?

An easier approach would be to load N (around 10 maybe) posts at the time and then just slide between them using jQuery, you could output an <a name="POST-ID"> before each one and use some jQuery sliding body plugin to slide to each <a>.

If you’d like to load via ajax or go to a totally new page with browser window refresh you could use the the function:

<?php
function get_adjacent_post( $in_same_cat = false, $excluded_categories="", $previous = true ){
        //from wp-includes/link-template.php line 1097
}
$next_post = get_adjacent_post( true, null, false );
?>

to get the next or previous post object which you wil get the post permalink by using:

<?php $permalink = get_permalink( $next_post->ID ); ?>

Hope that helps.