Chris89 Yes, this should be possible. Is this site on Beta 8.1? I'm going to assume it is for the purposes of these instructions, let me know if you're still on Beta 7.
What does your backup contain? Just the database, or also the files? The best ones to have would be your composer.lock file, composer.json, your config.php and the avatars from your assets folder. Pretty much everything else can be rebuilt (we have the technology!).
If you have the composer.json and lock files, drop them in your folder and run
composer install. If you have the config.php, you can add it in there too, and change the database fields to your new name and credentials. Change the URL field as well. If you don't have config.php, navigate to the folder on the web and install Flarum to the database (don't import your old one first, it will error).
If you don't have the composer files or config, that's okay. You can check the
extensions_enabled field of the
settings table in your database to see what extensions you had installed if you don't remember them all. Just install Flarum from scratch as per usual.
Then import your old database into the place you installed the new one, or pointed the config.php file to. The database doesn't contain any file paths themselves, though some extensions might need to be reconfigured with new URLs if necessary. So I'm pretty sure you won't encounter any problems, but let me know if you do.
If you have the avatars folder from the old one, you can drop that into the assets folder now.
php flarum cache:clear . Ensure that all of your extensions are installed and everything is working properly.
Let me know if you encounter any issues with this along the way, or as a result.