The Gatwick Express does NOT stop at East Croydon.
You don't say what time you have to check in at Gatwick, but assuming it's a couple of hours, you'll need to arrive by, say, 6am?
In that case, you will need to catch one of the Southern services from East Croydon. They take around 20-25 minutes. They are not terribly frequent that early on a Sunday, (one or two per hour) so you'll have to check directly with Nation Rail Enquiries (08457 48 49 50 from the UK, +44(0)20 7278 5240 from abroad). Also beware of engineering works on Sundays, which may affect services.
Bus services that early are very infrequent on a Sunday: there's a National Express bus from Coulsdon (3 miles south of Croydon) at 04.15 in the morning, arriving 04.38, which you have to pre-book... The train is a better bet in my view.