Omaha-One Night, Alone
I'll be in Omaha and Oklahoma City for one night each--a Monday and a Tuesday, respectively--in August, by myself, traveling by car through the Midwest. I like reasonably plain food, i.e., no fancy French sauces or Chef driven fancy plates. Fish or steak are good choices, but an exceptional burger or great sandwich selection will also work. Cost not an issue, but the choice must make a good martini and have a decent wine by the glass selection. Preference for a choice near a reasonable but clean and safe hotel.
Well, I'm not sure where you want to stay, but anywhere in Omaha is easily accessible by car. I know you said not fancy, but for great wine and dining experience, my friends rave about La Buvette. It's a small cafe/bistro with French food, but they have excellent wine and if the weather is nice you could sit on the patio and people watch in the Old Market. Fairly close is a Marriot, Hilton Garden Inn and a few others, though the Old Market is not too far from the cheaper airport hotels.
Another local joint with great food, especially Fish & Chips, is Dundee Dell. They also have bar food and sandwiches. It's a great neighborhood place and I have eaten dinner there alone and it was no big deal. They also have a great beer menu and a surprisingly excellent Scotch list. My guess is that they would have a great martini--not sure about their wine, though.
The Dundee Dell is close to the Element Hotel, which is a somewhat fancier place in mid-town. You may be able to get a really good rate there though. I stayed once last year and really enjoyed it.
Midtown also has J.Coco, a new restaurant opened by James Beard-nominated chef Jennifer Coco. http://jcocoomaha.com/ It has sandwiches on the menu and probably the best cocktail mixologist in town. It looks like they have a lot of great wines by the glass.
Hope you enjoy Nebraska! Let me know if you have any other questions or need any other assistance.