novice: options for data sync / mounts

Novice Linux user...  

I have two systems on different networks, only connected over the internet.  From system #1, I need a way to either mount a directory from system #2, or do something in a very efficient manner to keep data from a certain directory on system #1 in sync with system #2.  I do not have root on these boxes and cannot so something drastic like rsync.  Running a shell script from cron would be an option, but I don't want to have to download the entire directory every time I connect.. just the changes would be nice... so

1.  Can I create a mount across the internet between these boxes?

2.  Is there something like rsync that I could use to only replicate the changes?

Thanks in advance....