A celery cultivar, designated ADS-13, is disclosed. The invention relates to the seeds of celery cultivar ADS-13, to the plants of celery cultivar ADS-13 and to methods for producing a celery plant by crossing the cultivar ADS-13 with itself or another celery cultivar. The invention further relates to methods for producing a celery plant containing in its genetic material one or more transgenes and to the transgenic celery plants and plant parts produced by those methods. This invention also relates to celery cultivars or breeding cultivars and plant parts derived from celery cultivar ADS-13, to methods for producing other celery cultivars, lines or plant parts derived from celery cultivar ADS-13 and to the celery plants, varieties, and their parts derived from the use of those methods. The invention further relates to hybrid celery seeds, plants, and plant parts produced by crossing cultivar ADS-13 with another celery cultivar.