Yes. By default, WordPress will present the posts in order of newest to oldest so that part is already covered unless you’ve changed that behavior with a custom query.
In order to do this, you will need to create a new rewrite rule using the add_rewrite_rule() function that points /current-issue/ to the first post of that category/post type/taxonomy/etc that you are using to designate these posts. The resulting page should show only 1 post and have no reference to wp_link_pages() in the template file/part.
As for creating /previous-issues/, it will be the same principal but you’ll need a custom query with an offset of 1 in order to skip the first post.
Edit: One thing I forgot to add – after you add these rule you will need to flush the rewrite rules. This can be done simply by going to Settings -> Permalinks and simply saving them without making changes.