Free-space optical (FSO) systems are compulsory to realize high capacity and interference-free communication links from low-Earth orbit (LEO) satellite constellations as well as spacecraft and space stations to the Earth. To be integrated with high-capacity ground networks, the collected portion of the incident beam should be coupled into an optical fiber. To accurately evaluate the signal-to-noise ratio (SNR) and bit-error rate (BER) performance metrics, the probability density function (PDF) of fiber coupling efficiency (CE) must be determined. Previous studies have experimentally verified the CE PDF for a single-mode fiber, however, there is no such investigation for the CE PDF of a multi-mode fiber (MMF) in a LEO-to-ground FSO downlink. In this paper, for the first time, the CE PDF for a 200-μm MMF is experimentally investigated using data from an FSO downlink from the Small Optical Link for International Space Station (SOLISS) terminal to a 40-cm sub-aperture optical ground station (OGS) supported by a fine-tracking system. An average CE of 5.45 dB was also achieved given that the alignment between SOLISS and OGS was not optimal. In addition, using the angle-of-arrival (AoA) and received power data, the statistical characteristics such as channel coherence time, power spectral density, spectrogram, and PDFs of AoA, beam misalignments, and atmospheric turbulence-induced fluctuations are revealed and compared with the state-of-the-art theoretical background.