Hi there @Giorgos Dragoutas, thanks for posting on our Community and happy Monday!
You don't even need to upgrade your account to achieve this.
All you'd need to do is run the desktop app on both computers and make sure that this folder is inside your Dropbox folder. You'll also need to give enough time to the app to sync your changes, but those should automatically be reflected across your devices and the website as well.
I hope this helps!