As far as I know, Omaha is no different than the rest of the State if you hold a valid CHP.
I live west of Omaha and carry there often. When I first got my CHP I had to go downtown to OPD headquarters and register my EDC, they looked it over and made sure it wasn't stolen I guess!
That rule didn't last long. I see less and less "No Handgun signs" so watch for them.
Have fun, Greg