A lettuce cultivar, designated 50-0501020-B, is disclosed. The invention relates to the seeds of lettuce cultivar 50-0501020-B, to the plants of lettuce cultivar 50-0501020-B and to methods for producing a lettuce plant by crossing the cultivar 50-0501020-B with itself or another lettuce cultivar. The invention further relates to methods for producing a lettuce plant containing in its genetic material one or more transgenes and to the transgenic lettuce plants and plant parts produced by those methods. This invention also relates to lettuce cultivars or breeding cultivars and plant parts derived from lettuce cultivar 50-0501020-B, to methods for producing other lettuce cultivars, lines or plant parts derived from lettuce cultivar 50-0501020-B and to the lettuce plants, varieties, and their parts derived from the use of those methods. The invention further relates to hybrid lettuce seeds, plants, and plant parts produced by crossing cultivar 50-0501020-B with another lettuce cultivar.