Will a dermatologist be able to check a neck cyst for cancer?

Yes, often can: Yes but any surgeon can handle this problem if it is more than one cm in size.

Yes: A small superficial cyst on the neck is unlikely to be cancer. A much more likely cause is a sebaceous cyst. See: http://bit.ly/1yZnrMH

Possibly: Most cysts are removed by surgeons. See your doctor and get a referral in your area.

Start with PCP: It would be better to consult your primary care provider first and if needed s/he would be in a better position to choose the specialty for further consultation.