Calling $\mathbb{E}(d)$ the expected time for leaving the circle if you are at distance $d$, $\mathbb{E}(d)$ should have the following property: $$\mathbb{E}(d)=1+\frac{1}{2\pi}\int_0^{2\pi}\mathbb{E}(\sqrt{d^2+1-2d\cos\alpha})d\alpha$$ with the constraint $\mathbb{E}(d)=0$ if $d>3$.
This formula say that the expected leaving time is 1 (the cost of doing another step) + the mean of the expected time over each point at distance 1 from the current point ($\sqrt{d^2+1-2d\cos\alpha}$ is the cosine law applied at the triangle between the origin, the current point and next point).
I don't know how to explicit solve this equation (nor if is feasible to be done), but it can be used for iteratively approximate the solution.
edit: I've tried to solve with R, seem that the solution is $\mathbb{E}(0)\simeq11.42$, and this is the plot of the approximation of the function: !enter image description here