Can I use a neutral wire from another outlet to repair a broken neutral?
No, you cannot do this.  Assuming the nearby outlet you want to borrow neutral from is on a different circuit, you will be overloading its neutral.  Even if it is the same circuit, it's very bad practice to leave broken wiring energized without fixing it -- what if it moves a little and reconnects just enough to arc?  Then your house is on fire.  

However, fixing this properly probably isn't as much work as you think.  You most likely will not need to tear up your walls.  Unless you recently drove a nail through a wire, the wire inside the walls usually doesn't just break.  Rather, things usually go wrong at the outlets themselves.  The so-called "backstab" outlets are especially prone to having the wires come loose over time.  So the fix for this will involve pulling out each outlet in turn, making sure all screws, wirenuts, etc are tight, and ideally replacing any backstabs in the circuit.  Most likely you'll find one of the outlets had the white wire come loose, and when you tighten it, everything starts working again.