AirPlay isn't ideal because it's limited to 44.1 kHz / 16 bit. If you want to use listen to high-res material, you should use UPnP instead.
And you're right - if the Bricasti has UPnP capability, then it can already serve as a media renderer for playback. Buying an external streamer / server isn't necessary.
To stream to the DAC, you need to do a few things. The first is to set up a UPnP server. This can be done on your MacBook or directly on the NAS itself, if you go that direction.
I use a free and open source app called Kodi to enable UPnP on my Mac mini. Once installed, you need to add your iTunes Library as a music source. Then you need to enable UPnP on Kodi, including the options to share your music library, look for remote players (will need to show advanced options by click on the gear icon in the lower left of the navbar until it says 'Advanced'), and allow remote control via UPnP.
If you'd rather use a NAS such as Synology, there are packages available to enable UPnP functionality.
You can control playback and stream to your Bricasti DAC by using a UPnP app. For iOS devices, I suggest mconnect Player HD. The Bricasti should appear as an option when you tap the 'Play to' button the upper left. If you're on Android, I suggest BubbleUPnP. The Bricasti should show up as an option if you tap on the cast icon on the lower left.
Alternatively, you can also use the Kodi app interface directly by right clicking on a song and choosing "Play using". This is a bit clunky though.