Question: I will be arriving in Tbilisi on Tuesday 4 april at 04:30, and need to get to Baku from there as quickly as possible by bus.

My questions is, what bus station should I go to for transport at around this time?

Didube, Ortachala, Sadg. Moedani etc?

Any form of public transport is good - buses, mashrutkas, shared taxis, etc - so long as I can reliably find it at around that time in the morning.

Answer: 
My wife, native of Tbilisi, suggests that all buses and marshrutkas from Tbilisi to Baku leave from Ortachala. Search around Russian language sites seems to confirm it. (Note: all links below are to Russian language sites.)

[This site](http://travelask.ru/questions/55046-kak-dobratsya-iz-tbilisi-v-baku) talks about actually buses, not marshrutkas, that go from Ortachala to Baku.

[This site](http://za7gorami.ru/avtobusy_i_poezda_v_gruzii) specifically talks about marshrutkas and indicates that the route is from Ortachala to Baku, however this site states 2013 as the date.

[This site](http://permistika.ru/wiki/index.php?title=%D0%A2%D0%B1%D0%B8%D0%BB%D0%B8%D1%81%D0%B8) suggest a couple other options:

* marshrutka from main Railway station to "Red Bridge" (წიტელი ჰიდი - Tsiteli Hidi), then from there buses to Ganja and Baku
* bus from Lilo market near Tbilisi airport to Red Bridge, then buses as above
* marshrutka from main railway station to Marneuli, from there there are buses to Baku.

Finally, that same site suggests that you can take a taxi to the Georgia-Azerbaijan border, which costs about USD $40 ($10 per person, shared ride) - from there, cross the border on foot and take a bus from the Azerbaijani side.
Take a taxi to the border of Azerbaijan from the airport. Cross the border by foot and there are lots of buses and shared taxis waiting on the Azerbaijani side going to Baku.