One of the key difference between the upright bass and many of the smaller stringed instruments, is that the bass is not played with just the hands.

The strength to "fret", as well as pluck, the strings comes from the shoulders. The muscles of your shoulders and arms are stronger and have more endurance than the muscles in your hands. Instead of using your left hand alone to squeeze the strings into the fretboard, keep your elbow up and use your entire arm as a large "C clamp". This has the added advantage of keeping your left thumb and hand more mobile.

As for the right hand, especially for simple root-five bass lines, keep your plucking finger(s) almost parallel to the string, pluck with the meat (not the tip) of your finger, and simply let the weight of your arm, as it hangs from your shoulder, do the work. For faster runs or solos you can transition so that your right fingers are more perpendicular to the board, enabling more of a faster alternating finger technique, but for the meat and potatoes stuff let gravity do much of the work. (Even after three or four sets, gravity never seems to tire.)