  The ATLAS detector at CERN's LHC is preparing to take data from the first
proton-proton collisions expected in the next few months. We report on the
analysis of simulated data samples for production of heavy Quarkonium states
J/psi and Upsilon, corresponding to an integrated luminosity of 10 pb^-1 with
center of mass energy of 14 TeV expected at the early ATLAS data. We review
various aspects of prompt Quarkonium production at LHC: the accessible ranges
in transverse momentum and pseudorapidity, spin alignment of vector states,
separation of color octet and color singlet production mechanism and
feasibility of observing radiative decays Xi_c and Xi_b decays. Strategies of
various measurements are outlined and methods of separating promptly produced
J/psi and Upsilon mesons from various backgrounds are discussed.
