Linux equivalent to robocopy?

It sounds like rsync is definitely what you are after. You do not need to set up an 'rsync server' to copy files from one machine to another. Rsync supports copying files over SFTP (SSH File Transfer) which most linux boxes have enabled already (if not manually disabled).

See Lifehacker's Mirror files across systems with rsync for more details:

Whether you want to backup your data, distribute files securely or mirror your working documents over the internet from the office to home, between computers on your local network, or from your computer to your web server, rsync can get the job done. Today we'll use rsync to mirror folders between a Mac and PC over a secure connection at the command line.

Rsync is free (as in speech) and cross platform, meaning it syncs files between operating systems (Windows/Cygwin, Mac OS, Linux); it works over ssh so it's encrypted and secure; unlike FTP it's incremental, so only the parts of changed files are transferred, not whole files, which makes it go like Speedy Gonzalez; and the fact that it's command line makes it scriptable and easily automated.

rsync will copy files from one directory to another directory on the same machine like robocopy. Here is the rsync command that is equivalent to your robocopy command:

rsync -auv --exclude '.svn' --exclude '*.pyc' source destination

This will recursively copy the source to the destination and exclude older files as well as '.svn' and '*.pyc' directories/files.