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: 
Upon arrival in Tbilisi (04:30 in the morning) I went to the Courtyard Marriott hotel, whose staff has been super helpful to me in the past despite me not staying there. After doing some research, they said there was "unlikely" to be public transport to the border before 11:00

So they got me a taxi to the border for exactly 58 lari (USD 24 or EUR 22), saying that was a pretty good price.
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.