Do I need a DAC?

The odds are that it would. Whether it is significant enough or not will be what your ears decide.

I agree with Onemug. I wouldn't expect huge differences, but your amp and speakers should be revealing enough for you to hear a difference.

Your amp has a great feature that you may want to make use of. The amp and preamp sections of your NAD are connected by jumped rca connectors on the back. That means you can use it just as an amp or just as a preamp. If you were to get a DAC with a volume control, you can bypass the preamp section in the NAD. Doing it that way would probably make a much bigger difference. If you don't mind a little experimenting, you can try plugging your Airport Express directly into the amp section and use the volume control on your computer. You may be very surprised with the results.

You need DAC. MDHT labs tube DAC is a verryy good DAC only cost about $300-$500 on ebay.