The best way to get to Egelsbach without a car is by train.
S3 trains leave from Darmstadt main station at 5 and 35 minutes past the hour from 4:35 AM to 0:35 AM. (Terminus: Bad Soden) If you come in from Frankfurt, take the S3 from Frankfurt HBF (Main Station)/Hauptwache/Konstablerwache at 15/20/20 and 45/50/50 past the hour from 4:45 AM to 1:15 AM. If you're arriving at FRA Airport, take the S8/S9 (Terminus: Hanau HBF) or RE59 (quicker; Terminus: Frankfurt HBF) and change trains at one of the aforementioned stations.
Another possible (but way more expensive) way to get to Egelsbach is by plane. Egelsbach's airport EDFE is the largest airport of its kind (for smaller planes and helicopters) in Germany. However, flights may cost upwards of multiple thousand Euros per direction. If money isn't an issue, then this is a great and unique option, though scheduling may be required beforehand.