We present the discovery of a small kinematically decoupled core of 0.2$^{primeprime}$ (60 pc) in radius as well as an outflow jet in the archetypical AGN-starburst composite galaxy NGC 7130 from integral field data obtained with the adaptive optics-assisted MUSE-NFM instrument on the VLT. Correcting the already good natural seeing at the time of our science verification observations with the four-laser GALACSI AO system, we reach an unprecedented spatial resolution at optical wavelengths of around 0.15$^{primeprime}$. We confirm the existence of star-forming knots arranged in a ring of 0.58$^{primeprime}$ (185 pc) in radius around the nucleus, previously observed from UV and optical Hubble Space Telescope and CO(6-5) ALMA imaging. We determine the position of the nucleus as the location of a peak in gas velocity dispersion. A plume of material extends towards the NE from the nucleus until at least the edge of our field of view at 2$^{primeprime}$ (640 pc) radius which we interpret as an outflow jet originating in the AGN. The plume is not visible morphologically, but is clearly characterised in our data by emission-line ratios characteristic of AGN emission, enhanced gas velocity dispersion, and distinct non-circular gas velocities. Its orientation is roughly perpendicular to the line of nodes of the rotating host galaxy disc. A circumnuclear area of positive and negative velocities of 0.2$^{primeprime}$ in radius indicates a tiny inner disc, which can only be seen after combining the integral field spectroscopic capabilities of MUSE with adaptive optics.