Null-recurrence of a random walk
If $0$ were transient, then the total number $N$ of visits to $0$ is a geometric random variable with $p=\mathbb{P}_0(T_0=\infty)>0$ (probability of escape). That's because each excursion from $0$ is independent, with probability $p$ of successfully escaping. In particular, the expected number of visits is finite: $\mathbb{E}(N)=1/p<\infty$.
On the other hand,
$$\mathbb{E}(N)=\mathbb{E}\left(\sum_{n=0}^\infty 1_{(X_n=0)}\right)=\sum_{n=0}^\infty p_n(0,0)
=\sum_{n=0}^\infty {3n\choose n}\left({1\over 3}\right)^n \left({2\over 3}\right)^{2n}=\infty.$$
You can show this sum is infinite by using Stirling's formula to show that
$${3n\choose n}\left({1\over 3}\right)^n \left({2\over 3}\right)^{2n}\sim {c\over \sqrt{n}}.$$ Therefore, the state $0$ is not transient, so it is recurrent.
There are a number of ways to show that state $0$ is null. In your problem, put $x=y=0$ in (5.2) from Section 5.5 of Probability: Theory and Examples (2nd edition) by Richard Durrett to get:
$${1\over n}\sum_{m=1}^n p_m(0,0) \to {\mathbb{P}_0(T_0<\infty)\over \mathbb{E}_0(T_0)}.\tag{5.2}$$
Also $p_m(0,0)\to0$ as $m\to \infty$, implies that the left hand side in (5.2) goes to zero as well, hence $\mathbb{P}_0(T_0<\infty)/\mathbb{E}_0(T_0)=0$. Since $\mathbb{P}_0(T_0<\infty)>0$ we conclude that $\mathbb{E}_0(T_0)=\infty$.
Let's try to show $0$ is null-recurrent.
To get back to $0$, you have to do two $-1$s for every $+2$. So you have to take $3n$ steps, for some $n$, of which $n$ are $+2$s and $2n$ are $-1$s. The number of ways to do this is $3n\choose n$, and the probability of any one of these ways to get back to zero is $(1/3)^n(2/3)^{2n}$. So, now you know the probability of being at $0$ after $m$ steps.
I was going to say that this allows you to calculate the expected number of steps to get back to zero, but then I realized that some of the ways to get back after, say, $6$ steps include ways where you were already back after $3$ steps, so some adjustment in the formulas will be necessary to take this into account. So this is more of a suggestion as to how to proceed than it is a complete outline of a procedure.
Anyway, with some luck you'll be able to turn this into a proof that the expected time of return to $0$ is infinite, and then adjust the argument as necessary to show that every state is null-recurrent.