You get rsync dependencies with
[root]# ldd /usr/bin/rsync
libacl.so.1 => /lib64/libacl.so.1 (0x0000003aa8c00000)
libpopt.so.0 => /usr/lib64/libpopt.so.0 (0x0000003730c00000)
libc.so.6 => /lib64/libc.so.6 (0x000000372e800000)
libattr.so.1 => /lib64/libattr.so.1 (0x0000003aa8800000)
/lib64/ld-linux-x86-64.so.2 (0x000000372e400000)
Then you get package using yum:
[root]# yum provides /usr/lib64/libpopt.so.0
Importing additional filelist information
popt-1.10.2.3-34.el5.x86_64 : A C library for parsing command line parameters.
Repo : redhat
Matched from:
Filename : /usr/lib64/libpopt.so.0
So
yum reinstall popt
Should provide your dependenvy