How long have you got?! It's certainly possible by public transport, but not direct and it will take a long time. Buses run regularly from Matara,up the west coast to Colombo, from where you can get a bus to Hatton and, during the pilgrimage season, there are buses direct to Dalhousie.
There are also buses north from Matara to Pelmadulla or Ratnapura, which will take around 3.5 hours alone, and you then face another bus journey to Balangoda before looking for a local bus to take you to Dalhousie. If you have long enough and want to break the journey overnight, it would be easy enough to get to Haputale by bus from Matara (via the route mentioned above), then stay overnight and get the train to Hatton the next day, then bus or tuk-tuk to Dalhousie from there as we have done. The complete journey is very beautiful and varied and is well worth taking time over if you can spare it.
But if you just want to get there, hiring a driver is certainly your quickest and most straightforward option.
The Adams peak pilgrimage season starts from Dec On Poya day and end on wesak poyaday in May.
As mentioned by Susie if you have time overnight stay is advisable because after a long journey you have to start climbing which takes at lease 5-6 hours to complete the whole journey (climbing and getting down of Adam's peak).
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