Yes, you definitely can, but it will depend on a few things, such as whether it uses a database. If not, it's easier.
I used to do it a lot, and I would create a virtual site in /etc/apache2/sites-enabled/000-default.conf with the same name. Then I would create an entry in /etc/hosts to redefine that name to localhost. This part isn't really necessary, but it can be convenient if there are any absolute links. Then, I'd comment out the entry in the hosts file after uploading. But you can also create whatever name you want as long as it matches the name in apache's config file.
It's been a while since I've done this, but I may have a backup somewhere if you need specifics. But the basics are to add a VirtualHost entry after the existing one, something like this:
<VirtualHost *:80>
<Directory /var/www/html/MYSITE>
Options Indexes FollowSymLinks MultiViews
AllowOverride All
Order allow,deny
allow from all
</Directory>
DocumentRoot /var/www/html/MYSITE/
ServerName WHATEVER_YOU_WANT.X
</VirtualHost>