Of course you can OC as a non-resident. Remember, the laws don't grant permission, they restrict rights. Just be sure to carefully research all our silly ass laws.
Our Open Carry Flyer is a great place to start. It hasn't been given the official legal blessing of any attorney, judge, or jury... but I believe it is pretty accurate.
As for the cities you'll be travelling through: incorporation is pretty much the definition of a 'city.' Much like a firm may choose to incorporate, so do certain groups of people in one locality. These are known as cities. So, be sure not to get caught with a loaded firearm once you are in city limits.
Also, even among us here at OCDO, we can't agree on the interpretation of the statutes/case law on loaded open carry on public roads. I think that the intent of the legislature was that roads are not prohibited places, but I wouldn't want to be the test case. On any publicly owned road (which is pretty much all of them - I think I've seen 2 private roads in my 18 years here) you may want to cary unloaded.
Take a look at the flyer, and let us know if you have any other questions.