Suppose for the moment that we're talking about a SRW on $\mathbb{Z}^2$ and the target is the origin. Suppose that you're interested in the time interval $[0,T]$, meaning that you're asking about the distribution of the first hitting time of the origin, denoted $\tau$.

Then, very roughly, when you start at distance $r$ from the origin, the probability to hit the origin before getting to distance $\sqrt{T}$ from the origin is about $$\frac{\log(\sqrt{T})-\log(r)}{\log(\sqrt{T})} ,$$

in which case it takes about $T$ steps. So this is roughly the probability to not hit the origin before time $T$, when $T >> r^2$. For $T << r^2$ this probability is close to 1.

I hope this helps. It would be useful if you focused the question a bit more.

Suppose for the moment that we're talking about a SRW on $\mathbb{Z}^2$ and the target is the origin. Suppose that you're interested in the time interval $[0,T]$, meaning that you're asking about the distribution of the first hitting time of the origin, denoted $\tau$.

Then, very roughly, when you start at distance $r$ from the origin, the probability to hit the origin before getting to distance $\sqrt{T}$ from the origin is about $$\frac{\log(r)}{\log(\sqrt{T})} ,$$

in which case it takes about $T$ steps. So this is roughly the probability to not hit the origin before time $T$, when $T >> r^2$. For $T << r^2$ this probability is close to 1.

I hope this helps. It would be useful if you focused the question a bit more.