If that doesn't do the trick, I find that my tablet running Cyanogenmod 10.1.3 has rsync 3.0.7, but I've also installed extra bits and pieces ("Android Terminal Emulator" by Jack Palevich, "Terminal IDE" by Spartacus Rex, "Busybox Pro" by Stephen (stericson)) so I'm not 100% sure that it was originally available. You may be able to script and schedule something of your own based on scripts you use elsewhere, though with a few changes (e.g. "jping" instead of "ping", see Issue 29 on Terminal IDE's code.google.com page though that indicates that it may be fixed).
There's also a "lite" version, which only allows 2 accounts, no Tasker support and no sync filters (which I've never fiddled with anyway, so may not be that important).
There are two ways to write error-free programs; only the third one works.