We consider the possibility that the excess of cosmic rays near 1018 eV, reported by the AGASA and SUGAR groups from the direction of the Galactic Center, is caused by a young, very fast pulsar in the high density medium. The pulsar accelerates iron nuclei to energies 1020 eV, as postulated by the Galactic models for the origin of the highest energy cosmic rays. The iron nuclei, after about 1 yr since pulsar formation, leave the supernova envelope without energy losses and diffuse through the dense central region of the Galaxy. Some of them collide with the background matter creating neutrons (from desintegration of Fe), neutrinos and gamma-rays (in inelastic collisions). We suggest that neutrons produced at a specific time after the pulsar formation are responsible for the observed excess of cosmic rays at 1018 eV. From normalization of the calculated neutron flux to the one observed in the cosmic ray excess, we predict the neutrino and gamma-ray fluxes. It has been found that the 1 km2 neutrino detector of the IceCube type should detect from a few up to several events per year from the Galactic Center, depending on the parameters of the considered model. Also future systems of Cherenkov telescopes (CANGAROO III, HESS, VERITAS) should be able to observe 1 - 10 TeV gamma -rays from the Galactic Center if the pulsar was created inside a huge molecular cloud about 3-10* 103 yrs ago.