We present the physics and the antihydrogen production strategy of the AEGIS experiment at CERN. This strategy is based on a series of steps in which positronium (Ps), produced by e⁺ impinging on a porous target, is laser excited to high-n (Rydberg) levels and then made to interact with ultracold antiprotons (around 100 mK). An antihydrogen beam is then formed by Stark acceleration to be sent through a Moire' deflectometer to measure g for antimatter. The efficiency of the antihydrogen production process depends critically on the positronium excitation process which will be described in detail in the paper. The Ps cloud is produced within a relatively strong magnetic field at 1 T, with a consequent deep modification of Rydberg levels structure. A two-step laser light excitation is proposed, and the physics of the problem is discussed. We derive simple expressions giving the Ps excitation probability with feasible laser pulses suitably tailored in power and spectral bandwidth.