Yeast cells can be regarded as micron-sized and liquid-filled cylindrical shells. Owing to the rigid cell walls, yeast cells can bear compressive forces produced during the biotechnological process chain. However, when the compressive forces applied on the yeast go beyond a critical value, mechanical buckling will occur. Since the buckling of the yeast can change the networks in its cellular control, the experimental research of the buckling of the yeast has received considerable attention recently. In this paper, we apply a viscoelastic shell model to study the buckling of the yeast. Meanwhile, the turgor pressure in the yeast due to the internal liquid is taken into account as well. The governing equations are based on the first-order shear deformation theory. The critical axial compressive force in the phase space is obtained by the Laplace transformation, and the Bellman numerical inversion method is then applied to the analytical result to obtain the corresponding numerical results in the physical phase. The concepts of instantaneous critical buckling force, durable critical buckling force, and delay buckling are set up in this paper. And the effects of the transverse shear deformation and the turgor pressure on the buckling phenomena are also given. The numerical results show that the transverse shearing effect will decrease the instantaneous critical buckling force and the durable critical buckling force, while the turgor pressure will increase both of them.