Yes, the 81 goes into Castle Howard.
Little White Bus is just a normal bus- it'll be sat in the Station Yard when you arrive, at the south end of the southbound platform. No need to prebook unless you want a demand responsive service in between timetabled runs. There are enough timetabled runs that a day visitor is unlikely to ever use that facility.
You go down the ramp off the northbound platform, and under the bridge. Garsdale is a remote station so you can't possibly miss the bus. If you want to take a taxi back, or wait for one of the scheduled return trips.
If you get off the bus in Hardraw it's a beautiful, quite short river and field walk into Hawes. Then visit the Dales Countryside Museum and the Creamery. If short of time prioritise the creamery IMO. Get your fish and chips, then probably the last bus back.
But the Museum does a very good job of telling the history of the Dales.
The red squirrel extension needs to be pre-booked here- https://www.littlewhitebus.co.uk/red-squirrels/
All normal scheduled LWB trips are the standard national £3 flat rate bus fare.
Also at the north end of the southbound platform at Garsdale don't miss the statue of Ruswarp the dog- a touching story.
I think you kind of choose between Clapham with Ingleborough Cave OR Ingleton with the Nature Trail/Waterfall walk and White Scar Cave. One cave is probably good enough for most people. I would toss a coin on that. Whichever you choose pair with Kirkby Lonsdale-
When I wrote Ingleborough I of course meant Ingleton. Ingleborough is the hill/mountain above Clapham.
This is the 580/581 Timetable.
The bus changes number at Settle but you just stay on it. Likewise in Kirkby Lonsdale for anyone going all the way to Lancaster. It's just a legal issue, why they do that.
Rievaulx is really a day IMO although you may well be back in York in time for Evensong at the Minster.
For the Coastliner at York Station - the stop is called York Rail Station Stop RC- that is right outside the front door of the station, marginally to your left. All the stops are clearly signed. Take the 840 for Pickering- a 75 minute journey. Join the NYMR at Pickering. Most people do a round steam train trip, but you can do a single journey to Whitby then the bus back to York.
My suggestion would be single NYMR train trip to Whitby, time to explore the town then the 1745 bus back to York. The route is often voted as the most scenic in the UK- with every justification. Yes it's an hour back to Pickering or over 2 back to York.
But honestly it is worth every minute of that time. That ride is also the £3 flat fare.
The 843 will take you to Scarborough. Some people do make a day of it- 840 to Whitby, the 93 Whitby to Scarborough and the 843 home from Scarborough- but that isn't on your agenda.
You do have to book the NYMR.
Talking about Fish and Chips do visit Bizzie Lizzies (either branch) while in Skipton, also visit the Cheese stall on the market, and see if you can get an afternoon tea cruise on the Leeds and Liverpool Canal- https://www.canaltrips.co.uk/
and also Stanforth's butchers (on the hill just below the Parish Church) for their pork pies- a Skipton institution.