One-dimensional nanoribbons, thickness of which is smaller than width, exhibit special confinement and edge effects far from uniform in the cross-section. By means of density functional theory, we investigated the influence of the edges and doping positions on the electronic structure of Ag-doped ZnO armchair nanoribbons (ZnOANRs). Ag doping in monolayer and bilayer ZnOANRs (m- and b-ZnOANRs) have all been examined. The results indicated that there is no significant difference on the stability of Ag doping in different positions of both m- and b-ZnOANRs, but exhibits very different electronic properties directly related to the different doping positions. The depth of the acceptor states has essential relationship with the interaction between host O 2p and Ag 4d states in the acceptor. Ag substituting Zn atoms at the inner region of m-ZnOANRs could create shallow acceptors with small hole effective masses, benefit for p-type conduction. Ag doping in the inner region of b-ZnOANRs would create shallow acceptors but larger hole effective masses. The difficulty of hole mobility could be improved by increasing the Ag doping concentration in b-ZnOANRs. The discussion about the mechanism and the suggestion about the achievement in the experiment are interesting and timely.