See Ireland (5-6 days) hotel(s) or airbnb and best way to get from Dublin to York, England

What month?
How many is "we"?

Fly Dublin to Manchester then train to York from the airport

Ireland is alot bigger than it looks and 5-6 days is only 4-5 nights. I would suggest staying in Dublin the entire time. Bear in mind that each time you move to a different accommodation you would lose time checking out, checking in, packing, unpacking, etc. With 5-6 days (and please be clear whether those include arrival and departure days), you can take day trips to Newgrange/Knowth/Hill of Tara; Glendalough/Powerscourt/Avoca; Kilkenny; and Belfast. There would be little to no doubling back with those days trips Plus you still have Dublin to explore. Regarding where to stay, north of the River Liffey is generally less expensive than the area around Trinity College and St. Stephen's Green but still within walking distance. South of Trinity College/St. Stephen's Green is the Ballsbridge area. Also walkable to the principal tourist areas but slightly further out. How much you pay depends on location and time of year.

The other possible route to York is to fly to Newcastle then the metro to Newcastle Central and LNER or Transpennine Trains to York.
This is especially so if the trip is in March or early April when there are no trains from Manchester Airport to York due to the engineering work at Stalybridge.

It would be my husband and I, both seniors. He is not going to be able to walk very long.

Depending on the cost of the hotel or airbnb, we might be able to swing staying 6 or 7 days (not including travel days).

Thanks for any help or suggestions you can give us. We would rather see beautiful natural sites than go to touristy crowded places/towns.

We are planning on traveling to Ireland in the beginning of September and then meeting up with our friends in York, England after that.

On our most recent trip to Dublin we stayed at The Leeson Bridge Guesthouse located about a 10 minute level walk from St Stephens Green. Looks like their June rates are in the vicinity of USD$200 per night. There's an Express Bus stop right in front of the place for getting to and from the airport. Cost was about 7 euros last I checked.
You really don't need to move around during the short time you have. There's plenty to see in the city and there are organized day trips and tours available to take you north to Newgrange, Tara and the Boyne Valley or south to Powerscourt, Glendalough and the Wicklow Mtns.
You can also do independent day trips up to Belfast or down to Kilkenny - both of which are worth a day depending on your particular interest.
Along with flights to Manchester I see that RyanAir has direct flights from Dublin to Leeds, from which it's about a 25 minute train ride to York.
If extended walking is an issue I'd suggest booking one of the Hop On, Hop Off (HoHo) buses to give you an overview of the city and its attractions. Some have live commentary that can be both educational and entertaining if you happen to get a particularly glib driver. Selective use of the local taxis might save wear and tear on the legs too.

From Leeds Airport to Leeds Station (for York) you have to catch the Airport Flyer bus-
https://passenger-line-assets.s3.eu-west-1.amazonaws.com/transdevblazefield/YCD/A1-timetable-20221127-d96787d6.pdf
This stops right outside the front door of Leeds Station, and costs £4.

For completeness there is also an Airport Flyer bus to Harrogate where you can catch a train to York every 30 minutes- the more scenic route, and a quieter train ride as York to Leeds can get very busy. That bus costs £5.40.
https://passenger-line-assets.s3.eu-west-1.amazonaws.com/transdevblazefield/YCD/A2-timetable-20220724-c12ee556.pdf

But Leeds is the easier route if there are ambulatory issues, and you can pre book Assistance in boarding and leaving the train at Leeds and York if that is needed.

Or maybe your friends would be able to meet you at Leeds Airport.