This plugin creates lists of custom field information in the form of sidebar widgets.
How can I keep the individual Links during an plugin upgrade?
Don't deactivate the plugin. Because the options of the plugin in the options table of your weblog database are going to be removed automatically during the plugin deactivation. Further you have to upload the file of the new plugin version manually e.g. via FTP. Because during the automatic upgrade the old plugin would be deactivated.
Is it possible to use this widget with the K2 theme sidebar manager (plugin)? Yes, it is possible. Move the file custom_field_list_k2_widget.php to the /app/modules/-folder (e.g.: /wp-content/themes/k2/app/modules/) of the K2-theme if you are using the K2 theme with the old sidebar manager or the comparable plugin.
(Please, have look to "Other Notes" / "Usage" for further information, too.)