This paper is the first part of our study the asymptotic behavior of the solutions of the cylindrical Korteweg-de Vries equation (cKdV). In this first part, we consider the solutions from Schwarz's class and calculate the large time asymptotics using the method based on the asymptotic solution of the associated direct scattering problem. This approach was first suggested, in the cases of usual KdV and NLS equations, in the late 70s by Zakharov and Manakov. In a sequel to this paper, we plan to calculate the large time asymptotics of some other classes of solutions of the cKdV equation which exhibit the oscillatory type behavior, and we will also evaluate the short time asymptotics of the solutions of the cKdV equation. In the second part we will use the Defit-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems.