If you're looking for an interesting old Omaha restaurant, try Johnny's on 24th.
As St. George said, Old Fontennelle closed at least 15 years ago.
The Union Pacific Museum has nice displays of the Transcontinental Railroad era, but it has moved to Council Bluffs.
Durham Western Heritage Museum has various displays, but not necessarily western themed. Usually Omaha history, though.
Ft. Crook has several structures to tour that are from the Indian War era when Omaha was the headquarters of The Dept. of the Platte. (i.e., General Crook's quarters, etc.)
If want to travel a bit, up near Boyer's Chute north of Omaha a small ways, (a Lewis and Clark related site) you can visit the Bertram display, an 1865 steamboat sunk in the Missouri and excavated in the latter 20th century. It is literally a Time Capsule of the Civil War.
I notice you posted this 1:30 pm the day before you're expecting to be here, and I did not see this until I got home from work this afternoon. Oh well.
RCJ