Get the Ruin of Arnor sourcebook, if you can. I think it probably went out of print when Warbands were released (with RoA being rolled into Kingdoms of Men along with Rohan and Gondor). Its a great book, I reccomend it.
There are lots of ruined cities throughout Arnor. Fornost, Annuminas, etc. The Barrow Downs (burial mounds of the Dunedain) were a part of Arnor. Annuminas is a particularly interesting city - its a large city which was once the capitol (before the Capitol was moved to Fornost) on the shores of Lake Evendim, the largest lake in Middle Earth. Parts of it are even sinking into the lake, becoming flooded. The Weather Hills are a series of hills fortified with outposts and watchtowers (Weathertop/Amun Sul is one) along the border between Arthedain and Rhudaur (Rhudaur was the first Arnorian Kingdom to fall to Angmar, Arthedain the last).
If you have a half decent computer, I reccomend getting The Lord of The Rings Online. Its a great game and is Free to Play, though most of the game is restricted and must be purchased individually or you get it all (except the expansions) by subscribing. The North Downs (Fornost) and Evendim (Annuminas) must be purchased but the Barrow Downs can be explored for free. Oh, and the Shire is within Arnor as well - it was only settled by Hobbits in the later years of Arnor so you could do a Hobbit less version of the Shire (lots of pleasent green rolling hills, woods and fields. English countryside pretty much).The city of Fornost, ruined Capital of Arnor.