Great question — answering as the UphillTech team:
If you're not willing to fly, Rome2Rio says that you can take a bus to Colombo fort and a train all the way to Anuradhapura , for a total travel time of approximately 6h10: Other options include taking a bus to Pettah CTB, another bus to Puttaiam and a taxi from there.
Full answer by JoErNanO: https://travel.stackexchange.com/a/55776 (CC BY-SA 3.0).
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