Another Dublin airport hotel or stay in Dublin?

Look for a hotel on booking.com. Use the map view to find a suitable one close to the O'Connell St. Bridge in central Dublin. What's "real expensive" is in the eye of the buyer, but anything that central will cost more than anything farther out. After finding the right hotel, or narrowing the search, on booking.com, use the hotel's own website to make your reservation.

Then look at the routes and timetables for Aircoach. There's an express service, route 705X, between O'Connell St. and Belfast, via Dublin Airport. It runs hourly and takes a little less than three hours each way. That means you'll spend six hours of your day trip to Belfast sitting on the bus.

You might improve on this by spending one or both of your two nights in Belfast, depending on how that fits with the rest of your trip. Aircoach route 705X takes about two hours between DUB and Belfast. So you could go up there Friday or Saturday, spend both nights or just Saturday night, and come back to the airport Sunday morning with a time cushion before your afternoon flight.

Translink also run the competing X1 service on the same route- with some non stop X2 services between Belfast and Dublin airport (not serving Dublin City)-https://www.translink.co.uk/timetables?routeHashCode=de4f9090aba4b657dda5f80e3c8048ae9149ed527f305e58233241ac7a9ce106&transportMode=Bus&route=T0&direction=H

(the services marked as SE will not be running in October as SE means summer seasonal. Translink can be booked up to 28 days before travel.)

Sadly, hotels in Dublin are no longer inexpensive. No reason to stay near the airport for two nights especially with a late afternoon flight.

The train between Dublin and Belfast leaves from Connolly station in Dublin. If you are going roundtrip between the two, and not stopping at the airport, I'd suggest the train over the bus. It's more comfortable and you can get up and stretch your legs.

There are numerous bus lines between Dublin airport and city centre. Choose the one closest to your hotel.

https://www.dublinairport.com/to-from-the-airport/by-bus/dublin-buses

If you want to stay with Hilton, there is a Hilton Garden Inn in Dublin less than a 10 minute walk to Connolly station. I've stayed there and it's fine. They have a coin laundry which is great for travelers. But not cheap.

And therein lies the problem. I worked my whole life, often two jobs, to support my family and save for retirement. Now that I am retired and my income fixed, I want to enjoy my time and not be foolish with expenses. $300+ per night for a flop and a shower seems a bit much in the off season. Yes, nice lobbies and "Yes Sir", and "No Sir" up the wazoo are all well and good, but not for an extra $100 or so a night. I realize better location places pay more in taxes and can demand a higher price. I will pay more for the convenience. Double? Probably not.

I don't think I am cheap, just looking for what I consider a fair trade for a night's sleep. I am not partial to the hotel brand. I checked hotel.com, and for under $200 per night, only found hostels. I am closer to 100 years old than to 50. Hostels are not in my future

Have you tried looking at either of the 2 Premier Inn's in Dublin for that date? They seem to be coming in at around 200 Euro per night, or a bit more. It's the Saturday night which is the killer for you.
I looked at the Ibis (at the Red Cow) which is on one of the aircoach routes (as well as LUAS)- however they seem to be closed at the moment (probably housing refugees), and usually has good rates.
The PI at Temple Bar you would catch the LUAS (the tram) to and from Connolly and the Busaras
Check out is 12 noon, nice time for you to be at the airport by 1pm, 3 hours before your 4pm flight.
On each of the weekends the PI at the airport is far cheaper.
The Hilton is also 12 noon check out.
If the Airport PI is cheaper than the Hilton it may be worth switching, otherwise I think I would stick with what you have, have a lazy morning and accept the wait time.

Also Belfast (and especially Bangor) is coming in a lot cheaper than Dublin- way closer to your budget. Given the coach service frequency to DUB airport that may be worth a thought. Bangor may sound daft but there is fast and frequent train service to both Great Victoria Street/Europa Bus Station for the coach and Lanyon Place for the Enterprise train from Dublin. The PI is very close to the station. Yes I might have qualms about being in the wrong country- with a late afternoon flight, is it worth the trade off?