The ant's position at each time $n\in\mathbb{N}$ is a random variable which we could model in the complex plane as $Z_0=0$, $Z_n=Z_{n-1}+\Delta Z_n=\sum_{k-1}^n\Delta Z_k$ for moves $\Delta Z_n=e^{i\Theta_n}$ which are i.i.d. and which depend on uniform i.i.d. random variates $\Theta_n=2\pi\,U_n\sim\operatorname{Unif}\left(0,2\pi\right)$ . If we also let $ R_n=|Z_n|\,, \qquad S_n=\sup\limits_{1\le k\le n}R_n \qquad \text{and} \qquad T=\inf\left\{n\mid R_n>R\right\} $ be the first time $n$ for which $R_n>R=3$, then we are asking for $\mathbb{E}[T]$, which can be calculated as \mathbb{E}[T]=\sum_{n=R}^{\infty}n\cdot\mathbb{P}[R_{n-1} < R < R_n]\,. Now if we let F_n(r)=\mathbb{P}[R_n < r] and let $f_n(r)=\mathbb{P}[R_n = r]$ be the PDF and CDF of $R_n$, we can see that $F_n(n)=1$, and we may be able to determine these functions inductively, with a recursion: \eqalign{ F_n(r)&= \mathbb{P}[R_n < r]\\&= \mathbb{P}[R_{n-1} < r-1]+ \mathbb{P}[R_n The latter integral can certainly be split into the two cases s and $s>r$, for which the indeterminate probability inside the integrand can be evaluated using geometry by subtracting the area of sectors of circles (and triangles for $s>r$):
\eqalign{ \mathbb{P}[R_n where the angles $\alpha,\beta,\gamma$ can be expressed using the law of cosines: $ \eqalign{ \cos\alpha &=\frac{r^2+s^2-1}{2rs}\\ \cos\beta &=\frac{r^2-s^2-1}{2 s} \qquad\text{or}\qquad\sin\beta=r\,\sin\alpha\\ \cos\gamma &=\frac{-r^2+s^2+1}{2 s}=-\cos\beta\\ } $ (or solved from the equations $re^{i\alpha}=s+e^{i\beta}$ & $\beta+\gamma=\pi$, from which one can infer the diagrams for each case) and $A_\alpha=2\alpha-\sin2\alpha$ is twice the area of \{z\in\mathbb{C}:\,|z|>r,\,|z-s|<1\}.
Well, that's a start. There is surely a more elegant recursion; perhaps @carlop has found it.