It might take two weeks to build up some nice callouses, so stick with it. The first few weeks are always a little painful. Those callouses will make the ends of those fingers numb eventually. Take a short break whenever it starts to hurt too much. Look at the distance the strings are from the fingerboard. Some ukes need a saddle or nut adjustment. If the strings are too far from the fingerboard, your fingers have to press too hard (and it hurts). If the strings are too close to the fingerboard, the strings will buzz because they are vibrating against the frets. People who play vigorously need the strings to be high, while those who fingerpick more or strum less energetically can have the strings set up for "lower action".
–Lori