You do need a core. However if your NAS has an Intel processor it’s can act as that core. The core is always on, or at least it has to be on when using Roon. It also benefits greatly (stability and user experience) from being hard wired to Ethernet.
Check the processor on the Synology first. If it’s not up to spec then knowing where to get the Core software for the NAS is useless. Under spect Core will yield a poor result and turn you off from Roon. The laptop only needs to be on and running the Roon Core when you are using Roon. It need not be on 24/7.
Most Synology NAS models with a + on the model should work BUT your experience will be far from ideal based on Roon’s minimum specs. Certainly you can try, BUT again there are SSD requirements for the database location on the NAS too. I would try nd see how you like Roon on your laptop first, then think about either a NUC or other dedicated machine to run the Core on.
I started out using my laptop to run the Roon core. It only needs to be on when using Roon. It worked fine, but was a little awkward for me, so I bought a Nucleus to run the core. Using your laptop can get you started while you decide if you like Roon and want to invest in another device to run the Roon core.
(back to Crocs on my asymmetrical isolation feet!)