Wayland, you're on a winner with choosing Edinburgh. There are so much to do and see, only problem is that you may struggle to fit everything in for only a 2 day stay.

IMO, no need to look at airbnb...the likes of holiday express/ inn, ibis and travelodges offer a very decent stay and the beauty is that there are so many of them around central Edinburgh, you could easily pick up a bargain in Oct as its the off peak season here. Visit their websites and see for yourself.

I have lived in Edinburgh for a number of years and any time that I have visitors from afar who don't really know which places to go and what to do, i would take them to some of these: -

1/ Edinburgh castle- as mentioned above try to be there for the one o clock gun.
2/ Edinburgh zoo- you are fortunate to be here at the right time as in October, it is anticipated that the giant panda (i think called tin tin) on loan from China will due to give birth around this time...you cant believe the hype the locals have created!
3/ National Galleries of Scotland...its free
4/ Royal botanic gardens in Inverleith
5/ Helicopter ride over the city.
6/ Climb Arthur's seat in time to see the sunrise and enjoy the views.

If you are a golfer, then there are so many beautiful golf courses Edinburgh has to offer, all at very reasonable prices off course!

Hope it helps...