Recent angle-resolved photoemission (ARPES) studies of the La2-xSrxCuO4 (LSCO) compound provide a new piece of information on the evolution of the electronic structure in cuprate systems with doping. Our aim is to analyze that evolution by means of the spin polaron approach based on the string picture. We study the motion of a single mobile hole in the stripe phase of a doped antiferromagnet. Holes within the stripes are taken to be static, undoped antiferromagnetic domains between hole stripes are assumed to have an alternating staggered magnetization, as it has been suggested by neutron scattering experiments. The system is described by the t-t′-t′′-J model with realistic parameters and we compute the single particle spectral density. In this paper we concentrate on the filling level 1/12. Theoretical spectra and results of measurements performed at that hole concentration show reasonable agreement. In both cases we observe, along the direction (0,0)→(π,0), "splitting" of the spectrum and formation of new low energy states around (π,0). This splitting is also observed in the direction (0,π)→ (π,π). The spectra along the direction (π,0)→ (π,π) are much broader. That hinders us in assignment of any band for some range of k points in that direction. Finally, the spectra along the direction (0,0)→ (π,π) are very broad and it is also practically impossible to distinguish any sharp band.