I am the noncustodial parent and my ex is relocating but in the same city. Do I have the right to know her new address and phone number?
Generally, yes, you have the right to know where your child lives. This may also depend on what, if anything, is written in the parenting plan.
If there is no provision addressing it, she may not have to provide you with this information. She should provide it for legal reasons, but most importantly for the child's safety.
For example, if the child needed emergency medical care and the mother was unavailable (injured or unable to give consent), you could authorize the medical care.