The public transport in Brunei is not great. They have only buses and the buses start only at 11 AM. It is know for its cars as the cars are cheaper here. I did find a route map of Brunei public transport on the internet.
The above route map is of no use if you have to reach the airport before 11 AM. But note the bus fares are very cheap, it’s just Brunei $1 per person.
So the next option is taxi. Taxi from city to airport costs around Brunei $15-20. But the only problem with taxis are that there are only 50 taxis in Brunei and getting them is very difficult.
The government has introduced an application (mobile) called DART. The catch here is that you need to have a phone number to receive a PIN to activate the app. My Australian number struggled to work here, so I asked a local Brunei to help me activate and generally they are very friendly so they do help. Note: The taxi drivers only accept cash.
And the last option if nothing above works is to book a taxi online. For this option, the taxi has to be booked at least 24 hours before the actual time required.