Upvote:1
One option is by public taxi.
Rome2Rio doesn't cover everything (so there may well be other options), but according to Rome2Rio a public taxi is the only option that they're aware of, would take about 3.25 hours, and cost you in the realm of $55, assuming you aren't ripped off.
Upvote:2
We took a similar trip from Tissamaharama to Dalhousie with a private driver (we more or less drove past Udawalawe, so the answer might be applicable for you too).
Our host in Tissamaharama organized a private taxi for the trip. We were a group of three on that trip and paid around 5000 sri lankan rupees per person.
My advice: Ask your host if he knows a trustworthy driver.