Wow!!! What a beautiful building!!! Maybe not for chickens, though, they need dry and airy.
My understanding of a spring house is that it is essentially a root cellar - cool and damp in the summer and cool (not cold) in the winter. The damp is a problem for chickens. But if the spring inside is enclosed now or your region uses the term differently that might not be the case.
Ventilation might be difficult to add to a stone building. Chickens need about a square foot of open venting all the time: night and winter, too. And it matters where it is placed so you don't get drafts.