The use of low-dimensional objects in the field of cavity optomechanics is limited by their low scattering cross section compared with the size of the optical cavity mode. Fiber-based Fabry-Perot microcavities can feature tiny mode cross sections and still maintain a high finesse, boosting the light-matter interaction and thus enabling the sensitive detection of the displacement of minute objects. Here we present such an ultrasensitive microcavity setup with the highest finesse reported so far in loaded fiber cavities, F=195000. We are able to position-tune the static optomechanical coupling to a silicon nitride membrane stripe, reaching frequency pull parameters of up to G/2π=1GHz/nm. We also demonstrate radiation pressure backaction in the regime of an ultrahigh finesse up to F=165000.