Context: The space experiment CoRoT has recently detected transits by a hot Jupiter across the disc of an active G7V star (CoRoT-Exo-2a) that can be considered as a good proxy for the Sun at an age of approximately 0.5 Gyr. Aims: We present a spot modelling of the optical variability of the star during 142 days of uninterrupted observations performed by CoRoT with unprecedented photometric precision. Methods: We apply spot modelling approaches previously tested in the case of the Sun by modelling total solar irradiance variations, a good proxy for the optical flux variations of the Sun as a star. The best results in terms of mapping of the surface brightness inhomogeneities are obtained by means of maximum entropy regularized models. To model the light curve of CoRoT-Exo-2a, we take into account the photometric effects of both cool spots and solar-like faculae, adopting solar analogy. Results: Two active longitudes initially on opposite hemispheres are found on the photosphere of CoRoT-Exo-2a with a rotation period of 4.522 ± 0.024 days. Their separation changes by ≈80° during the time span of the observations. From this variation, a relative amplitude of the surface differential rotation lower than ~1 percent is estimated. Individual spots form within the active longitudes and show an angular velocity ~1 percent lower than that of the longitude pattern. The total spotted area shows a cyclic oscillation with a period of 28.9 ± 4.3 days, which is close to 10 times the synodic period of the planet as seen by the rotating active longitudes. We discuss the effects of solar-like faculae on our models, finding indications of a facular contribution to the optical flux variations of CoRoT-Exo-2a being significantly smaller than in the present Sun. Conclusions: The implications of such results for the internal rotation of CoRoT-Exo-2a are discussed, based on solar analogy. A possible magnetic star-planet interaction is suggested by the cyclic variation of the spotted area. Alternatively, the 28.9-d cycle may be related to Rossby-type waves propagating in the subphotospheric layers of the star. Based on observations obtained with CoRoT, a space project operated by the French Space Agency, CNES, with partecipation of the Science Programme of ESA, ESTEC/RSSD, Austria, Belgium, Brazil, Germany, and Spain.