Precise comparisons between the properties of matter and antimatter conjugatesconstitute a stringent test of CPT and Lorentz symmetries. The proton’s andantiproton’s magnetic moments have recently been measured to high precision inPenning traps, but further progress is impaired by the need to prepare a particlewith low motional energy. Current preparation schemes require long preparationtimes and are limited by high temperatures. Sympathetic laser cooling using anatomic ion has been proposed for preparation of low-energy protons and antipro-tons.This thesis presents the design and commissioning of a cryogenic Penningtrap system for sympathetic laser cooling using beryllium ions. The experimentaims to demonstrate direct Coulomb coupling between two particles trapped innearby, but separate potential wells in a Penning trap stack for the first time. Thistechnique could be used for sympathetic cooling of particles lacking the necessarysubstructure to apply laser cooling directly. The application of this method onprotons and antiprotons has the potential to decrease the mean kinetic energies ofthe particles and the preparation times required by several orders of magnitude.Furthermore, the method can be extended to other particles, such as highly chargedions.A quantum logic spectroscopy scheme for the measurement of the magneticmoment of the proton and antiproton has been proposed by Heinzen and Wineland.Experimental requirements for realisation of this proposal are discussed. The designof a suitable Penning trap system is described. A cryogenic ultra-high vacuumsystem cooled by a closed-cycle cryocooler, equipped with an ultra low vibrationinterface, is designed and commissioned. The necessary infrastructure, such as lasersystems and electronics are described.First signals taken from this newly constructed cryogenic Penning trap arepresented. Laser ablation trap loading, Doppler cooling and the reduction ofthe particle number down to a single ion are demonstrated. Prospects of theexperiment and implications for the precision of future measurements of the proton’sand antiproton’s magnetic moments augmented by sympathetic laser cooling andelements of quantum logic are discussed.