This thesis describes high precision measurements on the fundamental properties of the antiproton, namely the charge-to-mass ratio and the magnetic moment. This work is embedded in the experimental work of the BASE collaboration (Baryon Antibaryon Symmetry Experiment). BASE operates a sophisticated cryogenic Multi-Penning trap system in the Antiproton Decelerator facility at CERN. One main result of this thesis are significant technical improvements of the apparatus, which reduced the limitation of shot-to-shot cyclotron frequency scatter by a factor of more than five compared to earlier work.With this improved apparatus, a measurement campaign on the antiproton-to-proton charge-to-mass ratio with a statistical uncertainty of 20×10−12 and an overall uncertainty of about 35 × 10−12 was conducted. This campaign was part of a series of charge-to-mass ratio measurements on the antiproton with the overall goal to significantly improve the previous best measurement conducted by BASE in 2014, which yielded a fractional uncertainty of 69×10−12. In this thesis, also the first dedicated heating rate measurement in a cryogenic Penning trap experiment is described. Here, the lowest heating rates ever reported for an ion trap were observed. As part of this thesis, phase sensitive methods for measuring the cyclotron frequency of a single trapped ion were implemented in the BASE experiment. These methods allowed to measure the cyclotron frequency with a shot-to-shot scatter improved by a factor of five compared to methods used in previous experiments. The significant improvements in cyclotron frequency scatter open up the possibility to measure the antiproton magnetic moment and the antiproton charge-to-mass ratio with much increased precision, and thereby enable more stringent tests of the fundamental CPT symmetry by direct comparisons of matter/antimatter conjugates.