Is it flowing? If it is truly a natural spring then it reaches the surface alone. In that case, from what you say and the name of the place especially, pumping an ancient source may be difficult without permission. If it is a hand pump, one of the those big wheels or and 'up-down' lever action it would take a specialist metal worker to do it but a modern pump would require permission which could be tricky to get and disadvantageous. In either case it might be expense of little return. Springs dry up occasionally and land use can do that or divert them. Planting trees is one of the biggest offenders in fact.
Find out what you can about the place. Where I am, about 10 minutes walk to the Dordogne but then with a hydroelectricity plant from early 20 century at that place, it is now mainly forest. However, maps show a large area called 'les vieilles vignes' (not les vignes vieilles as one would expect), which describes what the main land use was up until the 1860s and 80s blights that killed off most vines in France. A look at local records show that the two hamlets here were once more prosperous and went into decline during the 20 century, intensive tobacco growing depleted some areas so that trees have been planted instead. The two local pumps, one well kept but dry, supplemented a stream just a couple of hundred yards downhill. That rarely dries up, especially being spring fed. So we can look for where water flows into that. Two springs we know of, one the former inhabitant of this house who is still a neighbour remembers, dried up as the forest grew. One went under our house and fed a large stone container with water. It is still there but never even gets damp. The other, in front of the house, is there somewhere, but probably deep although there is a place it bubbles up to the surface in front of our gate, destroying a patch of road surface after every storm or several days of considerable rain. I know more or less the exact route of the one that feeds our well and was worried when a house was built a few hundred metres away, but so far so good.
Once you get that kind of detail, often from long term local residents and a look at the old Cassini maps to show how it looked once, then considering the viability and purpose of the spring may make better sense. Bear in mind how many of them are more or less only there for some kind of historic sentimental reason rather than for a purpose and that restoration will inevitably cost an arm and a leg.