How effective are facet joint injections at relieving neck pain? I had a different type of injection in my lower back that didn't help.

Hard to say. Depends on your pain issue. Facet injections are helpful and can be long-lasting, however if your get short lived relief you may need to get evaluated for a radiofrequency ablation for longer lasting relief (greater than 1 year of relief is typical).
It varies. If your neck pain is coming from one or more bulging/herniated discs, then the injections may be quite beneficial. They are not as predictable for bony problems, such as arthritis or spinal stenosis.