The aerator on my faucet seems to have broken due to fatigue. As you can see in the picture the faucet is attached to the wall. The pipes run behind the wall. There are no shut off valves and I can't turn off the water without turning off all the building's water.
Can I replace the spout without turning off the water? If not, is there any way to attach an aerator to the unthreaded spout end?
If there was an aerator on it before the spout must have threads on it somewhere. Sometimes they are on the inside of the spout. You can replace the spout without turning the water off as long as you don't turn the handles. If that faucet is as old as it looks you might have trouble finding parts for it and it might be pretty difficult to take it apart. Good Luck.
If your spout pivots where it meets the main body, there may be athreaded coupling ther which you can loosen and remove the actual spout portion. If it comes off take it to a plumbing specialty shop that has been around a while.
If not, examine the aerator or what is left of it. If it is female and the thread of the spout itself is not damaged you may be able to soak it with a penetrant or lime buster and desolve corrosin and peel away the remaining aerator. If it is a male thread it will be a bit more tricky to remove from below and inside.