You can take either of the following, Anuradhapura to Kandy (AC) or Dambulla (public) bus, get off at Dambulla. Then from Dambulla to Sidiriya (public) bus.
Alternately From Dambulla take the (public) bus towards Habarana, get off at the junction for Sigiriya. Then you take tuk-tuk.
Same the other way round :
Many buses from Dambulla to Anuradhapura.