First, look for a cheap option that allows check-in at 12:30 AM, as the train goes to Lempuyangan.
The GoRide service isn't too far from Lempuyangan Station, and there are still plenty of GoRide services available at 12:30 AM. The GoRide fare is 15,000 Rupiah.
It's right next to Tugu, so it's close to the 6:00 AM departure time, so you can still squeeze in a short walk after dawn to get to Tugu. This is the best option for those looking for a cheap option, close to Lempuyangan Station and Tugu, and with check-in options according to arrival times.