In this paper, we investigate the nonlocal Kundu-nonlinear Schrödinger (Kundu-NLS) equation, which can be obtained from the reduction of the coupled Kundu-NLS system. Based on the analysis of the eigenfunctions, a Riemann-Hilbert problem is constructed to derive the N-soliton solutions of the coupled Kundu-NLS system. The N-soliton solutions of the nonlocal Kundu-NLS equation are then deduced with properly chosen symmetry relations on the scattering data. The dynamics of the solitons in the nonlocal Kundu-NLS equation are explored. The impact of the gauge function on the solitons is displayed for one-solitons. Compared with the dynamics of the two-solitons in the local Kundu-NLS equation, the two-solitons in the nonlocal Kundu-NLS equation display many differences. The repeated collapsing is a common feature of the singular solitons, and it seems that some of them are not the superposition of one-solitons. The singular solitons exhibit various behaviors in different eigenvalue configurations in the spectral space. Besides that, three kinds of bounded solutions are presented according to these eigenvalue configurations. In addition, two kinds of degenerate solutions are presented, and in particular, the positon solutions are discussed in detail. The decomposition of the positon solutions is analyzed and their trajectories are given approximately.