The first three steps on the guide for installing from precompiled binary on rclone.org/install is all you need:
curl -O http://downloads.rclone.org/rclone-current-linux-amd64.zip
unzip rclone-current-linux-amd64.zip
cd rclone-*-linux-amd64
You can do all of this without root access. The rest of the steps is for moving rclone to a place the system recognizes it as a command and installing the manpage. To move rclone to an environment path without root access use:
cp rclone ~/sbin/
This moves rclone to a enviroment path just for your user instead of the system, which is good because you’re on a public system. Finally, you don’t really need the manpage just use rclone --help or rclone.org