I believe pronouncing it like it's supposed to be pronounced in English is a safe bet. So, /nɑɹˈsɪsəs/.
I'm not sure where the japanese pronunciation might come from, but if I had to take a guess, it's either from Latin, where the "c" is /k/, or from the original Greek "νάρκισσος" /ˈnar.kis.sos/.
Taking the guess a bit further, it might be the case that they took the Latin nominative "narcissus" and removed the final -s, thinking that would get rid of the plural even if the plural of the second declension doesn't end in -s but in -ī, narcissī. It seems like a plausible misunderstanding to me, what do you think?
TL;DR:
The options seems to be either /nɑɹˈsɪsəs/ or /nɑɹˈsɪsə/ for the English pronunciation (introducing an /s/ for the heck of it in the first case), or if you don't mind stepping out of English's phonology, then you can go with the -incorrect- Latin /nar.ˈkis.su/.
In the end it's a made up word so whatever floats your boat I guess!
EDIT:
You can hear how the Latin word is supposed to be pronounced here: https://forvo.com/word/narcissus/#la
I decided to use IPA for this reply because trying to convey phonetic information without it, even more so when I'm not a native English speaker seems sketchy to say the least. I apologize if you are not familiar with it.