Is there a sin incurred by one on whose garment drops of urine have fallen, so he washed the spot, then performed ablution and prayed Dhuhr and Asr before washing the entire garment?

There is no sin upon you for what you did; rather, you performed what was obligatory by washing the area of impurity from the garment, and your prayer is valid. It is not necessary to change the garment or wash it completely; rather, it is sufficient to wash the area of impurity with water until the impurity is removed, and prayer in it will then be valid.