Many governments recommend against all travel to Iraq. See the Iraq article for more information.
Alqosh is an ethnically Assyrian town in the contested Nineveh Province of Iraq, but controlled by Iraqi Kurdistan.
Be aware that the road leading to Alqosh is littered with military checkpoints. For more info on how to deal with these checkpoints, see War zone safety. There is one route via a village called Sharya on which there will be less checkpoints and you will venture less deep into contested territory. However, if you get a taxi from Duhok, they will most likely take the road to Mossul and turn left into Zakho Road in the village of Badria. The taxi driver will know if this route is safe enough to travel on (it was in September 2022).
A return trip with a taxi from Duhok to Alqosh (plan some time for the town itself and its churches) as well as Rabban Hormzid might be as cheap as 40,000 dinars, but is likely to be more.
On the way back, you can get off in Qasara and hike up Zakho mountain to find the Halamata Cave with the Assyrian Reliefs.
2nd-order administrative division
Primary administrative division