The public transportation web page of Vaclav Havel Airport mentions various forms of transportation, some of which seem to operate only during day-time. However, according to the DPP web page, bus 510 operates at night-time from 00:15 to 05:00:
Na Beránku – I. P. Pavlova – Letiště (Prague Airport), bus 510
- interval: 30 minutes
- travel time: from I. P. Pavlova 45 minutes, total travel time 73 minutes
- fare: per PIT tariff,
- ticket on the 75 minute: 32 CZK
- all-day tiket (24 hours): 110 CZK
- three-day ticket (72 hours): 310 CZK
Since the bus runs every 30 minutes, and the average journey time is 45 minutes, you could catch the 03:00 bus, at the latest, to arrive at 04:00 to the airport. This of course depends on where your actually start your journey.
You could also book a taxi in advance. Once again, the Vaclav Havel web page on the matter comes in handy. Having done this in July 2015 I ended up spending 300
CZK (including tip) from Wenceslas Square to the airport at 04:00. My cab was prebooked and not hailed from the street. These prices are likely to become outdated very soon however so I suggest you call the cab company and ask for a quote.