We present the discovery of a small kinematically decoupled core of 0.″2 (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. We confirm the existence of star-forming knots arranged in a ring of 0.″58 (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″ (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 in radius indicates a tiny inner disc, which can only be seen after combining the integral field spectroscopic capabilities of MUSE with adaptive optics. Based on observations made at the European Southern Observatory using the Very Large Telescope under programme 60.A-9493(A).