How to transform Nondeterministic finite automaton (NFA) to regular expression equivalent
The Arden's rule, as it is usually stated, is easier to use if you consider equations on $(L_q)_{q\in Q}$ depicting the language $L_q$ the automaton accepts from state $q$. Doing this, you obtain the following equations. Check that you understand this properly :


$L_0 = 0L_0 + 1L_0 + 0L_1$
$L_1 = 0L_3 + 1L_3$
$L_3 = \varepsilon$


(I use $L$ instead of $q$ as it looks less misleading to me) 

Once you have those equations, you can solve this as follows


$L_1 = 0 + 1$ (I replaced $L_3$ with its value)
$L_0 = (0+1)L_0 + 0(0+1)$ (Replaced $L_1$ and factorized. This is ready for Arden's rule)
Since $\epsilon\not\in L_0$, from Arden's rule : $L_0 = (0+1)^*0(0+1)$


The language accepted by the automaton is always the union of the $L_{q_i}$, where the $q_i$ are the initial states. So here, $L = L_0 = (0+1)^*0(0+1)$