= Solution
Let $h_i=\mathbb E_iT_0$ be the <expected hitting time> of state $0$. For $i\geq1$, <first-step analysis> gives
$$
q^{-(i+2)}(h_{i+1}-h_i)
+q^{-i}(h_{i-1}-h_i)=-1.
$$
With $d_i=h_i-h_{i-1}$ this becomes
$$
d_{i+1}-q^2d_i=-q^{i+2}.
$$
The minimal nonnegative solution has
$$
d_i=\frac{q^{i+1}}{q-1},
$$
so
$$
\boxed{\mathbb E_1T_0=h_1=d_1=\frac{q^2}{q-1}}.
$$
Equivalently, <Kac's lemma> gives $\mathbb E_0T_0^+=1/\pi_0$. A first step from $0$ either returns immediately with probability $1-p$ or moves to $1$ with probability $p$, so
$$
\frac1{\pi_0}=1+p\,\mathbb E_1T_0.
$$
Substituting $\pi_0=(q-1)/(q-1+q^2p)$ gives the same result.
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