Any good tips about how to travel in rural areas where train travel is not an option -- and without renting a car? Is there an alternative besides Uber or Lyft?
Buses are the usual method between towns of moderate size, though they may not drop you at actual rural locations. See www.buseirann.ie and www.translink.co.uk. Regular taxis also exist.
Why are you assuming there will Uber and Lyft? Did you research?
Without a car or usage of a train you'll need to rely on busses for transport. Local ones are infrequent. Takes some scheduling and can eat up hours of time. Cross county ones are good.
Good Luck