Solution

Initial state is:

$q_0$

From $q_0$, using $\epsilon$ move:

$q_0 \xrightarrow{\epsilon} q_1$

From $q_1$, again using $\epsilon$ move:

$q_1 \xrightarrow{\epsilon} q_2$

So, without reading any input symbol, we can reach:

$q_2$

But final state is:

$q_3$

Now from $q_2$, on input symbol $a$:

$q_2 \xrightarrow{a} {q_2,q_3}$

So, after reading one $a$, we can reach final state $q_3$.

Therefore, the minimum string is:

$a$

Length of $a$ is:

$1$