How to put out the fire of an eternal burning artifact on a water limited map?

Fire is hot enough to evaporate some water before it can be extinguished, so the main problem here is getting enough water. There are two ways you can gather water.

You've mentioned murky pools in your embark area. Murky pools will refill when it rains, so if you're not in an arid biome you can dig channels from your pools to an artificial reservoir, block them with floodgates, and empty them into the reservoir after rain pools up in them.

Alternatively, caverns are known to have large underground lakes in them, which will continuously refill if they touch the edge of the map. With a system of pumps or a well and much patience, you can bring this water to the surface, and eventually have enough to extinguish the artifact, or for other needs.

Don't add one unit each time. Just dig 4x4 area near the block containing fire. Connect it to reservoir(murky pool). Connect to to fired artifact with a flood gate. (This is important)Block the flood gate.

When the 4x4 reservoir fills completely, open the flood gate.

The excess water will guarantee to extinguish the flame.

According to the wiki, water should extinguish the artifact. Since the source of ignition was dragonfire, the temperature might be very high and will need time to cool down.