reverse the order of characters in a string

I know you said "without third-party tools", but sometimes a tool is just too obviously the right one, plus it's installed on most Linux systems by default:

[madhatta@risby tmp]$ echo 12345 | rev
54321

See rev's man page for more.

Simple:

var="12345"
copy=${var}

len=${#copy}
for((i=$len-1;i>=0;i--)); do rev="$rev${copy:$i:1}"; done

echo "var: $var, rev: $rev"

Output:

$ bash rev
var: 12345, rev: 54321