Sleight,
I took several buses during my visit to Sicily a few years ago so I can understand what you may be going through trying to find the various bus schedules, connections, etc.
If you’re set on staying in Noto (instead of Siracusa), you can take the Interbus from Palermo to Noto with a change in Siracusa, and thereby avoiding Catania Airport.
The bus station in Palermo is behind and to the left of the train station (Palermo Centrale)
You can buy your ticket(s) the same day of travel but arrive early just in case there’s a line.
I would suggest buying both tickets (Palermo to Siracusa, and Siracusa to Noto) in Palermo.
It’s possible that it may just be one ticket with a transfer since you plan to travel to Noto the same day.
I’d definitely ask if the bus to Noto will wait if the first bus is delayed. Both buses will be Interbus (blue colored bus with large Interbus logo on its side)
The buses have a designated parking area with good signage at the Palermo bus depot (outdoors with some benches)
You’ll need to place your luggage in the hold below before boarding the bus, and the bus driver will check your ticket as you board the bus. Take your personal bag with you.
The bus “station” in Siracusa is just along a street (Corso Umberto- just google Siracusa bus station for map and photos) there is no station building per se. There’s only a small kiosk where one can buy tickets.
(I took the Interbus to Noto for a day trip)
Looking at the bus schedules, it looks like you’ll only have 5 minutes to change buses when you arrive to Siracusa, but the buses will be at the same stop so that shouldn’t be a problem.
The only issue I can foresee is that if the first bus (Palermo to Siracusa) is delayed, I’m not sure the bus going to Noto will wait.
Now, if you end up choosing the bus to Catania Airport, you’ll transfer to the AST (Azienda Siciliana Transporti) bus at the same location.
As someone else mentioned, there is no bus station. All buses arrive and depart from the same area. When I was there (2018), there were a few kiosks for the various bus companies.
I don’t recall any benches there, and you must be very vigilant and make sure you’re ready to board when your bus arrives.
It can be a bit chaotic because several buses may arrive and depart at the same time. In my experience, if the bus drives up and another bus is parked in its designated “stop”, one will need to get to the bus, the bus will not come to you.
Enjoy your time in beautiful Sicilia!
Edited to add…
If you opt to stay in Ortigia (Siracusa), there’s a small bus (navetta) that runs from the bus stop to Ortigia. When I was there, the navetta driver was grumpy and told me that I couldn’t ride the navetta with my luggage, but then made an exception. You’ll need to buy your ticket (€1 in 2018) from a yellow machine located behind the driver.