We asymptotically calculate the spatially averaged mean first passage time (MFPT) of a diffusing channel protein in a finite membrane patch containing a small absorbing anchor site. Different two-dimensional membrane geometries are considered including a circular, a square-shaped, a rectangular, and a cylindrical domain. The asymptotic expressions are found to be in excellent agreement with results from Monte Carlo simulations if the radius of the diffusing protein is sufficiently small. For a larger radius, a simple correction to the asymptotic expressions is proposed. We show that the average MFPT for a circle and a square-shaped domain of the same area are approximately equal as long as the anchor site is close to the center of the domain. We also discuss how the average MFPT depends on the aspect ratio of a rectangular and a cylindrical domain. Among such domains with a fixed area, a minimal MFPT is obtained for the square-shaped domain.