Can I use a neutral from another circuit to install a 3-way switch?
No, you should not do this.  Think of what would happen if someone wanted to work on that other circuit - they would flip the breaker and assume its off, yet there is still current on the neutral coming from another circuit!  Zap!  Even worse, someone disconnects the neutral without knowledge that it used elsewhere and now you have an open neutral!

As well, it's worth mentioning that the neutral has no phase - they are all tied together (and tied to ground at the main panel).

The correct way to do this is to run a new cable.