Right across the street from you, Ahmad's is definitely one of my favorite Omaha restaurants. The Fessun Jun (pomegranate chicken) is my favorite dish there. If you call a day or two ahead of time he'll even make it for you with lamb instead of chicken. I agree with sladeums that VMertz is also a great choice (it's within a block of your hotel too). They have a great wine list.
there's not really a lot of ethnic restaurants downtown.
these couple are the best:
Ahmad's Persian Cuisine
1006 Howard St
Great food and service. Prices are a bit high for my tastes, but I would say it's generally worth it. Reservations strongly recommended.
A menu is available on this site:
1010 Howard St
Generally solid Indian food. Great service. Reservations highly recommended for dinner.
and then there's this place, which i have never been to:
The International Cafe
2416 Farnam St.
African restaurant adjoining a grocery store. One of a few restaurants in town that serves goat. This review from the local paper a couple years ago will give you the general idea:
These two (non-ethnic) places are also downtown and generally considered to be two of the better restaurants in town:
1722 Saint Marys Ave
Upper mid scale. Solid food. Menu on their site. Reservations strongly recommended.
V Mertz Restaurant
1022 Howard St
Upscale. Pricy and classy. The setting in the Underground Passageway is really nice. Menu on their site. Reservations strongly recommended.
for breakfast take a hike up the street from the hotel to:
Big Daddy's Diner
1901 Farnam St
Diner atmosphere. Stupidly huge portions, breakfast all day long. Great homemade food.