so... I'm not a tech guy, but when I've got 50 $ ticket to use on digital ocean, I've tried my best to do it, but there is a problem. I don't have nor domain, neither a website. If i will create a free wordpress website and after attach it to my domain, can i attach already attached domain free wordpress site to a digital ocean 10 $ hosting?
You should move the domain over to Digital Ocean, (it would make this process easier to explain and guide you through)
Then you need to export your MySQL database (http://webcheatsheet.com/sql/mysql_backup_restore.php)
And then move all your files over to a new server on Digital Ocean (https://codex.wordpress.org/Moving_WordPress)
This would definitely be a project, there is no easy "attach to digital ocean" it would be a move to a new server for sure.
I would recommend, recreating the site on a Wordpress Droplet, and if there is a lot of content, work on moving that over to the new database.