Travel from Westport to Leenane

Transport for Ireland is your friend here for all public transport in Ireland

Bus Eireann bus #423 at 0710, 0910, 1155, 1345, 1615 and 1830 , 45 minute ride 7 days a week

Assuming you don't have a car, it appears that Bus Eirann operates route 423 between Westport and Clifden with a stop in Leenane. Here is the timetable:
https://www.buseireann.ie/routes-and-timetables/423

I followed your instructions, but when you look at the timetables, the dates skip from Oct-5 to Oct-28> What is that about?

If you look at any other service on the Bus Eireann website it does the same thing in jumping dates. It just reflects that the Irish know there is a bus and don't book aeons ahead, if it all.
The late October dates showing are the autumn school break dates, so services will vary as any route with a school day variance will subtly differ. Hence why they show.

In many ways the Transport for Ireland website referenced above is the national public transport reference tool.
It correctly gives the service as running daily, at least as far as Christmas. I haven't looked further out.

Note you want the Leenane in County Galway, not the ones in County Cork or County Tipperary!