Add a “Next Post” & “Previous Post” styled button manually to a post

This is standard WordPress functionality, contained in previous_post_link and next_post_link. You don’t need to edit functions.php. Just add to your template:

<div class="buttons">
  <span class="previous-button"><?php previous_post_link() ?></span>
  <span class="next-button"><?php next_post_link() ?></span>
</div>

Now you can style the buttons with css. Follow the codex links for customizing options to the functions.