 Solution

State $2$ can be reached from state $1$ only at an even time. In the two-step sampled chain, each step while still at state $1$ reaches state $2$ with probability $4/9$, and otherwise stays at state $1$ with probability $5/9$. Thus, if $T$ is the original hitting time,Equivalently $T=2G$ where $G$ has the geometric distribution on positive integers with parameter $4/9$.