Pests
Psila rosae: Use of physical crop covers (nets) is recommended in small scale production.
Diseases
Sclerotinia and other soil borne diseases: Use of natural enemies for control (Coniothyrium minitans) where available
Storage diseases should be managed by regulation of hygiene aspects (harvest, handling and storage condition)
Virus: see aphid control for all apiaceous crops.
Specifically for celery: Celery Mosaic Virus (CeMV) requires the largest possible crop rotation intervals (see also $. 2 and 3).
Seed treatments against diseases are recommended.
Weeds
Preferred options are: Mechanical weeding between rows, band treatments, (false/stale seed beds).