I've recently been put in charge of a Ubuntu server in the Linode platform.
This webserver is running a production Rails application with a MySQL database.
I need to ssh into it to download a copy of the database to perform further development locally.
I've read that Linode requires to shut down the node before letting you reset the root password from the rescue tab in the manager. Is the "Shut Down" button in the dashboard safe to use and restart a production webserver? Won't the server lose any data? (I'm asking this because I'm used to Amazon AWS, where servers are not persistent)
Is there other way to access the server via SSH within the Linode Manager?
Linode offer an alternative path into linode VMs, which is the linode shell (aka lish). Documentation is here. However, you'll still need the root password, or another username and password, to actually log into the system.
To answer your other question, linodes are persistent. If you shut the system down safely (which the shutdown button in the linode dashboard will do), it is unlikely you'll lose any data.
I say unlikely, but there is some chance that the previous admin set up some ramdisk-backed data, or has not commmited some changes to disk, or some other thing that may get lost, however in the absence of existing documentation or a root password, you can't work these things out anyway.