popen gives you a file handle on a process’ stdout, not its stderr. Its first argument is interpreted as a shell command, so you can do redirections in it:
FILE *p = popen("prog 2>&1", "r");
or, if you don’t want the stdout at all,
FILE *p = popen("prog 2>&1 >/dev/null", "r");
(Any other file besides /dev/null is acceptable as well.)