The Ellersley House is really close to the airport bus route but to be fair is more than a couple of block from the city centre (unless you are doing American measurement) it a decent 25 min walk from the west end of town let alone old town.

A lot of this will depend on your budget, There are lots of chain hotels which should be cheap and available but cab get busy at weekends. I might suggest the IBIS which is just off the mid point of high street, or Jurys in Jeffrey street. There also Premier inn Travel Lodge and apex hotels. If your only here one night I imagine you want to be out more than in the hotel. although hotels in the grassmarket are noisy at weekends. Even the royal scot in abercrombie terrace is a possibility very nice location in georgian terrace in the new town not sure about the price for that tho. You could also try hotels in royal terrace at the east end of the city centre but so you know its on the edge of the pink triangle

Quite a lot of rental companies here will drop cars, just be aware parking at many of these hotels is limited.