Neck xray came back fine. But still have very mild neck pain and tingling in my fingers. Will this go away on its own? I'm 16 and also have anxiety.

Wait another week. Many herniated discs like what sounds like what you are having will resolve on its own in a few days to weeks. If you are still in a lot of pain and anti-inflammatories are not helping or if you notice numbness/weakness presenting itself then seek medical care to evaluate this further.
See details. This may simply be an anxiety reaction. However, have it further evaluated to be certain. A neurologist may help and an EMG of the upper extremities would also determine if nerves were impaired.