An inversion algorithm to estimate the discharge of rivers observed by the forthcoming SWOT mission (wide swath altimetry) is developed and assessed in detail. The algorithm relies on an advanced variational data assimilation formulation applied to the Saint-Venant equations (1D shallow-water) combined with a lower complexity model (low Froude assumption and locally stationary). This modeling approach enables to estimate from the altimetry measurements the three flow features: the discharge Q(t) associated with an effective bathymetry b(x) and a (non constant) roughness coefficient K. The river geometry is built up with effective cross sections defined from the altimetry measurements. The numerical results are analyzed for three rivers (\sim100 km long each) presenting rapid flow variations compared to the observation frequency. Two scenarios of observation are considered: frequent satellite overpasses corresponding to the SWOT Cal-Val orbit (\sim1 day period) and SWOT like data corresponding to a \sim21 days period with 1 to 4 passes at mid-latitudes. Given prior mean values (of Q or/and b), the numerical experiments demonstrate that the inference of the river discharge Q(t) and the bathymetry profile b(x) may be accurate. Various prior values sources are investigated in view of worldwide applications. Once the assimilation of a long period of measurements is done (e.g. one year long time series), the low complexity flow model is correctly calibrated and provides accurate discharge estimations in real-time from newly acquired measurements.