Sabiha Gökçen is the Asian-side airport at Kurtköy, Pendik; Istanbul Airport (IST) is 90 km away on the European side. They are different airports with different coach companies: HAVABÜS runs from SAW (havabus.com), Havaist runs from IST (hava.ist). The names are one letter apart, and a ticket for the wrong one is worthless.
Kadıköy is the terminus and it is on the Asian side. If your hotel is in Taksim, Sultanahmet, Galata or Beşiktaş you need a second leg - a ferry from Kadıköy, or the Marmaray - which adds about ₺75-95 and takes the total journey to 90-120 minutes. The metro is cheap; it is not one-seat.
The M4 runs roughly 06:00 to 00:00, with later services on Friday and Saturday nights only. HAVABÜS's own timetable runs from the airport to 01:30, with extra coaches between 01:00 and 04:00 for delayed flights. A 1am landing on a Tuesday leaves the coach or the taxi.
HAVABÜS is a private coach, not a city bus, so the İstanbulkart transfer discount that applies on the metro and İETT routes does not apply to it: you pay the full coach fare with no reduction for the connection at the other end. Bank cards and cash work at the desk and on board.
HAVABÜS has moved its Sakarya (Adapazarı) departure point: those coaches now leave from Sabiha Gökçen metro station exit 1, not from the terminal. The change is published on the operator's own site, and it is the kind of detail that only bites when you are already outside with a case.