Finite time hitting probabilities for Brownian motion in the plane

Instead of a concrete answer, I will give what appears to be the most useful reference. I quote the first paragraph of Wendel, J. G. "Hitting spheres with Brownian motion". Ann. Prob. 8, 164 (1980).

Let $X_t$ be a standard $d$-dimensional Brownian motion with nonrandom starting point $X_0$. When $d \ge 2$ we seek explicit formulas which will determine the joint distributions of the first time $T \le \infty$ and place $X_T$ (which is only defined when $T$ is finite) where $X_T$ hits a sphere centered at the origin, either from the inside or from the outside, or exits from the region bounded by concentric spheres.

See also Betz, C. and Gzyl, H. "Hitting spheres from the exterior". Ann. Prob. 22, 177 (1994) and various cites of these papers.