This work presents an analytical solution for electro-osmotic flow (EOF) in small amplitude oscillatory shear (SAOS) as a measuring tool suitable to characterize the linear viscoelastic properties of non-Newtonian fluids in microchannel flow. The flow in the straight microchannel is driven by applying oscillating sinusoidal electric potentials. Fourier series are used to derive an expression for the velocity field, under an externally imposed generic potential field aimed at the practical application of SAOS in characterizing the rheological properties of viscoelastic fluids. This extensive investigation covers a wide range of parameters and considers the multi-mode upper-convected Maxwell (UCM) model, which represents the rheology of viscoelastic fluids in the limit of small and slow deformations. Particular focus is given to two cases of practical interest: equal wall zeta potentials at both channel walls and negligible zeta potential at one of the walls. The results show that in flows with thin electric double layers (EDL), Reynolds numbers below 0.001, and Deborah numbers below 100, corresponding to a viscoelastic Mach number of 0.32, the velocity field outside the EDLs is linear and has a large enough amplitude of oscillation, which may allow the quantification of the storage and loss moduli in the linear regime. This technique requires the use of significantly smaller sample sizes than the traditional SAOS in a rotational rheometer.