  Let D be a smooth relatively compact and strictly J-pseudoconvex domain in a
four dimensional almost complex manifold (M,J). We give sharp estimates of the
Kobayashi metric. Our approach is based on an asymptotic quantitative
description of both the domain D and the almost complex structure J near a
boundary point. Following Z.M.Balogh and M.Bonk, these sharp estimates provide
the Gromov hyperbolicity of the domain D.
