I just did a quick search and a taxi is about $15 to the city. I flew into Noi Bai a few years ago and took a share taxi (van) to the city. It was $2/person, leaves when full. I told them what hotel and they took me there. Of course they try to steer you into a hotel where they get a kickback, but I told them I had a reservation and wanted that hotel. There are scams going on at Noi Bai. Google 'scams noi bai'. Be aware of them.
I walked out of the airport and asked several touts where the share taxi was located. Found it after some false info from taxi drivers. It was a girl with a clipboard herding visitors towards a van. There is also a VN Airlines van that goes to the Old Quarter, $2-3. Jetstar Air & Kumho also run buses to the city, figure $2-4. This might help.
Another option: I would chat with other English speakers while in line for Immigration or getting your bags. Find others going to the Old Quarter. Decide what hotel is best located as your destination. All of you get out at that hotel. The $15 is for the taxi, NOT per person. Split 3-4 ways, this is a decent option into the city. It will be less than $1 for a pedicab or moto to your Old Quarter hotel. You will not need VN dong to pay for the taxi or hotel. They will take USD. Once in the city, get over to a bank and change money. You can change money at the airport if you feel you need VND.
Good luck with your planning.