Solution:
The following is such a string for $n=19$ :
$$9|18|7|16|5|14|3|12|1|10|11|2|13|4|15|6|17|8|19$$
where the vertical bars indicate breaks between the numbers. On the other hand, to see that $n=19$ is the minimum, notice that only one digit can occur an odd number of times in a palindromic $n$-string (namely the center digit). If $n \leq 9$, then (say) the digits $1, 2$ each appear once in any $n$-string, so we cannot have a palindrome. If $10 \leq n \leq 18$, then $0, 9$ each appear once, and we again cannot have a palindrome. So $19$ is the smallest possible $n$.